Michael Eriksson's Blog

A Swede in Germany

Posts Tagged ‘math

A few thoughts on educationrealist

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In December, I read large portions of the blog educationrealist.* I found it particularly gratifying that the author (henceforth “Ed”) verifies a great number of my opinions on schools and schooling with “from the trenches” information regarding current U.S. schools.**

*Already briefly mentioned during a recent blogroll update. I wrote most of the below a few weeks before publication, based on keywords and short descriptions gathered in December. Taking up writing again today, I can no longer recall much of what I had intended to write for the remaining keywords. This has led to some points being considerably more abbreviated than others. I was torn between throwing them out altogether and keeping the short version, but mostly opted for the short version. With hindsight, I should also have kept more links.

*My opinions are based on a mixture of my own experiences from Swedish schools in the 1980s and early 1990s, reasoning from principles (of e.g. human behavior and abilities), less detailed accounts by students or teachers, and discussions by (mostly) other outsiders. Correspondingly, there was a risk that the non-trivial changes over time or when moving from country to country had mislead me. This does not appear to be the case.

Among the interesting observations to be made:

  1. There is a strong component of innate ability to school success.

    This has corollaries, many contrary to what politicians tend to believe, like: It is not possible to teach everyone everything with a reasonable effort. A one-size-fits-all* school system will fail many students through under- or over-challenging them and through necessitating pedagogical compromises. Over-education is wasteful and unproductive at best. Ignoring group differences in “academic talent” is a recipe for failure.**

    *Ed usually discusses this in terms of (absence of) “tracking”, which is one way to make the school system “multi-sized”. I note that during my own school years more-or-less no such efforts of any kind took place. Cf. e.g. some discussion of skipping grades/being held back in [1]. No in-year acceleration or other differentiation, from which I could have benefited greatly, were available to the gifted. The first true differentiation took place in (the rough equivalent of) senior high-school, where students self-selected into more specialized programs based on interest, with some minor filtering based on previous grades when there were more applicants than places.

    **This especially with an eye on racial variety (which was almost a non-issue during my own school years, with an almost homogeneous population). Many posts deal with racial realism, the evils of various affirmative action measures, etc., approaching the statistics driven topics of “The Bell-Curve” from a more practical/personal/anecdotal angle. However, in the big picture, this is not limited to race—I note e.g. how German news-papers and politicians ever again complain about how the German system would hinder working-class children, without even considering the possibility that the differences in outcome could be partially caused by differences in (inherited) abilities that affect the respective probability of the parents being working-class and of the children doing poorly in school.

  2. The grade system is broken through rewarding effort, compliance, whatnot over actual ability and performance. Indeed, the picture painted is much bleaker than during my own school years, where there was a strong subjective component in the teacher’s evaluation, but where, at least, performance was measured through tests—not home work.

    This is particularly interesting in light of an earlier text on admission criteria, where I oppose the suggestion to remove Högskoleprovet (“Swedish SATs”) for admissions to higher education in favor of a purely GPA based admission.* If we assume that the same trend is (or will be) followed in Sweden, the correct resolution would be to abolish GPA admission and rely solely on Högskoleprovet… (But just as Ed complains about the dumbing-down of the SATs, there is reason to fear that Högskoleprovet is suffering a similar faith. There certainly is a constant fiddling with it—notably, to ensure that boys do not outscore girls.)

    *Swedish admissions are centralized and use numerical criteria—not interviews, essays, extra-curriculars, …

  3. The negative effects of destructive students on others can be considerable.

    Interesting sub-items to consider is what type and degree of disciplinary measures should be allowed, and the benefit of splitting students into groups that are more homogeneous in terms of e.g. interest and behavior. (Yes, the latter might make it even worse for the trouble students, but they are not exactly thriving anyway—and doing so would improve the opportunities for everyone else.)

    I did some minor reading on this from other sources (but did not keep links), and found some stories that make even Ed’s experiences, already well beyond my own,* look harmless—including a female teacher writing about regularly crying with frustration in the evening…

    *To speculate on the difference, I note that I spent a fair bit of my school years in small classes, that anti-authority attitudes were not yet as wide-spread, and that Ed has taught many classes of a remedial nature. Racial factors might also play in, e.g. in that the cognitive differences in the class-room are greater in the U.S. or that many minority boys have a deliberate “tough” image. I know too little of his situation and experiences to say anything with certainty, however.

  4. Student motivation is highly important, and often something that the school system fails at (but which is often blamed on the student).

    This is the more depressing, seeing that a knee-jerk political reaction to school issues is to increase the time spent in school, which obviously will reduce motivation further even among the motivated, let alone the unmotivated. It also comes with other problems. Someone fails in school due to lack of motivation? Put him in summer school so that he will enter the following year already “school tired”. Let him repeat a year to prolong the torture. Let him take remedial classes to make his days longer. Etc.

    The correct solution is, obviously, to attack the lack of motivation (which is very often to blame on the school/teacher/school-system/… in the first place). If this problem cannot be fixed, other efforts are pointless or even harmful. If it can be fixed, the strong students will advance on their own, weaker will at least have a chance, and we have to have enough realism to be willing to part with the too weak students at an earlier time than “year twelve”.

  5. Politicians and education reformers are often very naive.
  6. There is a lot of trickery with re-classification of children, artificial passes of courses, and similar, for the purpose of making schools look good (or “not disastrously bad”?).

    A particularly interesting variation is the confusion of classes for/students in “English Language Learning/er” and special education: Apparently, many students who should be in special ed are put into ELL based on excuses, e.g. because the parents were first generation immigrants, while the child is a reasonably proficient native speaker who happens to do poorly in school. This way, the failure in school can no longer be blamed on the school (or, God forbid, the possibility that not all students are equally smart)—but on an alleged language handicap.

A point where his experiences (and some citations?) do not match my expectation is the competence level of teachers: He repeatedly expresses the view that the effect of increasing the subject* competence levels or minimum test-scores** of teachers has little effect on student outcomes. There is even some speculation on a negative effect on Black students, because they appear to do better with a Black teacher, and increasing the test-score limits would reduce the proportion of Black teachers. My own experiences with teacher competence are very different, but I could see a possible reconciliation in teachers affecting different students differently, e.g. in that a dumber teacher will bore/under-challenge/annoy/whatnot the bright students, while a brighter teacher might similarly over-challenge or have troubles with adapting to the dumber students—leaving the total effect on the student population roughly constant. (Similar explanations could include e.g. brighter teachers being stricter on dumber students when grading than dumber teachers are, resp. dumber teachers failing to appreciate good answers from brighter students.***) If this is so, we have an additional argument for segregation by ability (combined with corresponding choices of teachers); while ignoring teacher competence would be particularly bad for the brighter students.

*E.g. requiring better math knowledge in a math teacher. This in contrast to e.g. pedagogical training, where I am uncertain what his stance is—apart from a negative opinion of some of the training actually on offer.

**On some type of qualification test for teachers. Similar statements might or might not have been made concerning e.g. SAT scores or GPA.

***With several of my own less bright teachers, what I said sometimes went well over their heads. More generally, I have made the life-experience that stupid people often are under the misapprehension that someone brighter disagrees because he lacks insights that they have, while the true cause is typically the exact opposite—he has insights that they lack.

Looking at Ed, himself, he appears to do a great deal of experimentation and tries to improve his teaching over time. There are a few things that appear to work well for him and that could prove valuable elsewhere, including (big picture) running a hard line against students, treating students very differently depending on their behaviors/need/abilities/…, and attempts to motivate his students, as well as (on the detail level) many pedagogical tricks and techniques.

Unfortunately, there are a few other things that strike me as negative, even if some of them might be a result of external circumstances, e.g. that the school system leaves him with no good options or that he must make compromises between the interests of the students, his school, society, whatnot. This applies especially to his “D for effort” policy, which makes him a contributor to problems that he, himself, complains about, e.g. misleading grades and remedial students making it to college (while still being remedial). My take? It is never “D for effort”, it is never “E for effort”, it is absolutely never, ever “A for effort”: Unless actual accomplishment results from the effort, it must be “F for effort”. (Which, to boot, makes for a phonetically better saying.)

Another negative is a considerable mathematical naivete for a math teacher,* that is likely the cause of some weird ideas that are more likely to hinder than help his students, e.g. that higher order polynomials (or functions, depending on perspective) are arrived at by “multiplication” of lines** (i.e. first-degree relations like y = 5x + 3). Yes, this is a possible perspective, but it is just a small piece of the overall puzzle, and it strikes me as highly counter-intuitive and pedagogically unsound as an approach. (In my preliminary notes, I have a second example of “identifying numbers graphically only”, but I am not certain what I meant. It might have been something like requesting students to draw a graph and find the y-value from the x-value by measurement, instead of calculation, which would be pointless as an “only”, but could be acceptable as a preliminary step or to demonstrate the occasional need to use other methods than pure calculation.)

*In all fairness, he, unlike many others, understands and acknowledges that his understanding is superficial when he moves beyond the classes that he teaches.

**Generally, there is an extreme over-focus on geometry; however, I am not certain whether this is caused by Ed or the school (or the text-book publishers, politicians, whatnot). This includes e.g. viewing functions more-or-less solely as graphs, root learning of sine and cosine values, and similar.

Yet another is “lying to students” (see excursion), as demonstrated e.g. in a post on “The Evolution of Equals”. This post also shows some examples of enormous efforts being put in to teach the trivial to the dumber students, who might not belong in high school to begin with—at least a basic grasp of the equals sign should be present years earlier. Move them out of school or to some more practical course and use the freed teacher resources to teach those teachable… (Some other posts make a better job of displaying a great effort with little return, but this is the one post for which I kept the URL.)

Some other points could be seen as positive or negative depending on the details. For instance, he does some type of interactive/quizzing teaching that expects a “chorus answer” from the class. This might keep the students alert and force them to at least rote-learn some material—but it does not allow for much true thought and it does not demonstrate any deeper understanding among the students. I would certainly have found it annoying (or worse), had it been applied during my own school years.

Excursion on a generic solution to tracking, acceleration, etc.:
I have for some time considered taking a more “collegey” approach to school as a solution (sketch) to some problems. I see some support for this in the non-integrated approach taken to e.g. math in Ed’s descriptions.* What if the material to be covered, even in year one, is broken into rough packages of four quarter-semesters per semester and topic—and the students then go through these packages in whatever tempo they can manage? The strong students will soon move ahead of schedule, be it in general or in their favorite topics. Similarly, the student with an interest in a certain area, e.g. math, can move ahead in that area. The weaker students can take their time until they have mastered the matter sufficiently well. Etc. Exactly how to handle the teachers in this scenario is not yet clear to me, but it is clear that mere lecturing** to the class would have to be considerably reduced or combined with a division of people based on the package that they are currently involved with.

*Math was integrated through-out my own school years. While I do not see this as a pedagogical problem, it does limit flexibility.

**With some reservations for the first few years, I consider lecturing to be highly inefficient, often boring, and increasingly only suitable for weak students as we move up in grades. Strong students are able to learn mostly on their own and based on books. Cf. an earlier text on college material. In at least a U.S. context, it also helps with hiding the problem of sub-grade-level literacy—better to reveal and address the problem.

Excursion on memory:
A recurring issue is that Ed’s weaker students often actually do learn how to do something—but have forgotten it again by the next semester. This is likely partially caused by a too superficial understanding,* but it could also point to many simply having very weak long-term memories. Revisiting some past interactions with others, such a weak memory could explain quite a few incidents that I had hitherto considered rooted in e.g. an original pretended understanding or agreement,** willful non-compliance using pretended ignorance as an excuse, too great a stupidity to be able to make even a trivial generalization of a known fact, or similar. (Whether weak memory is the explanation I leave unstated, but it is something that I must consider in the future.) A twist is that I have partially not considered memory an issue, because I thought my own memory poor and rarely had such problems—but in comparison to some of Ed’s students, my memory is excellent…

*Understanding does not only help with recollection, but can also be used to fill in many “blanks”. Of course, in terms of school, it can require a teacher with the right attitude: I recall an oral examination (on the master level, no less) where the professor asked for a formula. I had not bothered to learn the formula, knowing that the derivation was very easy from first principles, and set about deriving the formula. He immediately interrupted me, stating that he was content with the formula and that the derivation was out of scope. Apparently, he expected students to blindly memorize the formula, while having no clue how it came about…

**Something that also occurs among some of Ed’s students, as might some of the other items mentioned.

Excursion on lying to students:
“Lying to students” roughly refers to giving them a simplified (or even outright incorrect) view, which is (perceived as) good enough for now and which they can easily understand—without telling them that it is a simplified view. The result of this is that those who do not progress in their studies believe things that are not true, while those who do progress have to unlearn and relearn things in a highly unnecessary manner. A particular complication is that it can be very hard to be certain what opinions/knowledge/whatnot, gathered over a prolonged time period, corresponds to what state of knowledge. In many cases, the simplifications can make something harder to understand for the bright students, because it simply does not make sense or because the non-simplified version is (in some sense) cleaner. A very good example is the theory of relativity taught on the premise that the speed of light in vacuum is fixed* vs the premise that there is an upper speed-limit on causality or information, which light reaches in vacuum—the latter is much easier to see as plausible, leads to more natural conclusions, etc.** To toy with a simpler example in Ed’s direction: Compare the teacher who says “It is not possible to subtract a larger number from a smaller number!” with the colleague who says “If one subtracts a larger number from a smaller number, the result is a negative number—but that is for next semester!”. Which of the two is more likely to have confused students the next semester? Possibly, to the point that other claims made are no longer seen as credible? Which is more likely to peak an interest into what negative numbers are? Possibly, to the point that ambitious students read ahead or ask for explanations in advance?

*In all fairness, this could be based less on a wish to (over-)simplify and more on historical development. Even so, it should not be the starting point today.

**Consider e.g. questions like “What is so special about light?!?”, “Why must it be the speed in vacuum?”, “What happens when light travels through a crystal at a lower speed?”, …

Written by michaeleriksson

January 14, 2019 at 10:42 am

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A critical look at PISA

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A few weeks ago, I downloaded a PDF with sample questions from the 2012 PISA math test*; today, finally got around to look at it.

*Linked to and discussed in some article somewhere. I do not remember the details.

I find myself being highly critical, with my main beef being the excessive amounts of irrelevant text, and the associated lack of abstraction and clarity. Consider e.g. the first problem group (“MEMORY STICK”) with formulations like:

Ivan has a memory stick that stores music and photos.


Ivan wants to transfer a photo album of 350 MB onto his memory stick


During the following weeks, Ivan deletes some photos and music, but also adds new files of photos and music.


His brother gives him a new memory stick with a capacity of 2GB (2000 MB), which is totally empty. Ivan transfers the content of his old memory stick onto the new one.


Not only are such formulations patronizing, more-or-less calling the test taker a child to his face, but they and the unduly concrete formulations distract from the actual math, hide the math, and introduce a too large aspect of reading comprehension*: A math test should test math ability—not reading comprehension**. This in particular when it comes to a test that could put students under time or other pressure, where the translation from text to math could prove to be a stress factor for many of them. To boot, there is at least a risk that the results are misleading through blending out the ability to handle abstract problems. “2 = 2 = ?” is a math problem; “Jack has two cookies and Jill has two cookies. How many cookies do they have in sum***?” is not.

*Likely also other irrelevant factors relating to the translation from text to math.

**I similarly recall once buying a book with mathematical and similar puzzles, likely by Martin Gardner, and ending up throwing it away: Not because the puzzles were to hard, but because I had to waste too much time wading through a sea of text to isolate the handful of data that actual was relevant to the respectively problem—boring and without an intellectual challenge. Only afterwards could I focus on solving the problem, which was what I wanted to do. This is very much like trying to watch a DVD and finding that the actual movie cannot be started before a number of copyright warnings, mandatory trailers, animated menus, …, have wasted several minutes of the viewer’s time.

***As an aside, I saw a similar formulation in a different context, for a younger audience, but using “[…] do they both have”. This is a good example of how incompetent question makers can ruin a question: The expected-by-the-question-maker answer would be four; the correct answer in the most reasonable textual interpretation is zero—there are no cookies that they both have.

Of course, there are many instances where a corresponding translation is needed in a practical situation; however, such translations are mostly not very hard and they tend to differ from the textual for at least two reasons: Firstly, in a practical situation the problem solver picks the relevant facts out of the practical situation—not out of a text by someone else describing the practical situation. To boot, the texts for “math” problems like these tend to not describe practical situations—just theoretical situations someone has translated into practical terms in a simplistic manner. Secondly, the view of a practical situation can often make aspects of the problem, thought errors*, unexpected complications, whatnot, obvious that are not so in a text.

*A good example of such obvious thought errors is one of the few problems I got wrong: “The ice-cream shop”, question 3. The question requires placing sets of chairs and tables within a shaded area, under a constraint regarding the walls of the surrounding room. Being in too much hurry, I just focused on the shaded area without considering that the walls did not coincide with its borders. This error would, admittedly, have been easy to avoid, had I taken my time—but it would have been virtually impossible to commit when standing in the physical room. This type of textual problem differs in quality from a real-life problems (to more than the roughest approximation), in a manner similar to how e.g. racing a car in a computer game differs from doing so in real life.

An added disadvantage of these text-heavy problems is “cultural loading [bias, whatnot]”*: The text introduces opportunities for such problems that would otherwise not be present, especially in light of potentially suboptimal translations (also cf. below).

*I am normally skeptical to complaints in this area, seeing that e.g. I.Q. tests tend to be abstract; that cultural knowledge tends to lower differences between groups, through adding an irrelevant factor; and that the cultural difference from test taker to test taker is usually comparatively low to begin with. Here we have a test intended for extensive global use, where little or no effort has been put in eliminating cultural variations, where there is an additional severe translation complication—and where the very point of the test is to compare and evaluate different countries! (Whereas e.g. I.Q. tests are conceived to compare and evaluate individuals.)

Some more specific criticisms:

  1. A few the items come with translation notes (the document being intended more for test makers and test administrators than test takers). However, there is typically no obvious reason why a specific point has a translation note and so many others do not. Worse, the translation notes are often highly specific, e.g. referring to translation into French (but not German or Swahili)*. To me, these notes mostly serve as a proof that the test is suboptimal.

    *For instance, `Translation Note: In French, “penguin” is “manchot”.’ Do they consider specifically French translators to be idiots? Is there some (unmentioned) odd complication around penguins in French? (If so, are there really no other language with the same problem?) Of course, if the questions had been made abstract, there would be no need to mention penguins in any language…

  2. There are quite a few unfortunate formulations that could lead to unnecessary errors—and one where the formulation is outright incorrect: “Question 4: MP3 PLAYERS” states “The normal selling price of the MP3 items includes a profit of 37.5%.”, which would normally mean that 37.5% of the overall price is the profit. However, what is actually meant is that the price includes a mark-up, not a profit, of 37.5%. It is true that a later sentence claims “The profit is calculated as a percentage of the wholesale price.”, referring to the same profit; however, in combination, this is an extremely non-standard usage and in order to take this into consideration, the reader basically has to ignore the fact that he has a clear claim. A reasonably analogy would be a question claiming “a gin-and-tonic includes 37.5% gin” and then slapping on a “the percentage is relative the amount of tonic”. To boot, even a careful reader would not necessarily make the corresponding modification, because it would be equally conceivable that the several uses of “profit” referred to different concepts*. (This was another question I got “wrong”; however, unlike with the “ice-cream shop”, I put the blame on the test makers.)

    *E.g. in a scenario of “Given the profit (as a percentage of the selling price), give the profit (as a percentage of the wholesale price).”, incidentally showing that it would be better to use “profit” for the amount only, and otherwise speak of e.g. “profit margin”.

  3. “CHARTS” uses a poorly structured and hard-to-read diagram* as data input. Coloring, spacing, and lining contribute to introducing an entirely unnecessary complication; it can even be disputed whether this type of diagram was suitable for the data at hand**. Being able to read a diagram is a valuable skill, but here it is not just a matter of understanding how to read data from the diagram in principle—there is also an optical complication that made my eyes water.

    *Generally, the examples using some type of excel-style diagrams give an argument that such diagrams are more-often-than-not inferior to a table with the same data: Save diagrams for complex data where the visual can truly help in detecting trends and connections—do not throw them together willy-nilly because “diagrams are cool”.

    **A bar chart; a line chart would would likely have been more appropriate. It can also be disputed whether it really made sense to combine all four entities in one chart, or whether one chart per entity would have been better. (Assuming that we do not use a proper table to begin with…)

  4. At least question 5 of “CHARTS” is ambiguous through the talk of a “trend” that “continues”: When we speak of a continuation, it is the question what continues. Here we deal with diminishing CD sales, and in a real-life scenario, it would be highly likely that a continuing trend would be measured by a percentage (e.g. sales diminishing by twenty percent per month) or otherwise be measured relative the remaining sales; however, looking at the previous data, from which the extrapolation must be made, it appears to be more of a fix drop. (The instructions for the test administrator do indeed speak of a “linear trend”.) When extrapolating a trend, however, a model is needed, and it is highly simplistic to just assume e.g. a linear trend—even when a handful of data points point towards an approximately linear relationship. There are other models that might match the data, especially when factoring in the risk of a diagram distorting the data ever so slightly.*

    *Indeed, using my original numerical approach, with approximate read-outs, I repeatedly landed above 400 (compared to 370 as the allegedly correct answer), on at least one occasion close to 500. (Note that this is still close enough that I would have picked the right option from the multiple-choice entries.) Only after using a knife to approximate a straight line from A to B did I find 370 acceptable. However, even this is approximate, because I had to guess where the crossing line for July was… (Note: I am unaware of the equipment available to the test takers. If graded rulers are allowed, better “measurements” are possible, and correspondingly better outputs are to be expected—but at a cost of boring detail work that would have been unnecessary, had the test makers had the common sense to use a table of data instead of a diagram…

  5. “Question 2: PENGUINS” is extremely naively modeled and/or poorly formulated, to the point that a bright* student could** get caught up in time-consuming speculation about the correct-yet-unrealistic assumptions to make. The hitch lies in “By the end of the year 20% of all the penguins (adults and chicks) will die.”: The eventually needed model assumes that the deaths will all occur at the end of the year (or at least after the other main event of the year, the raising of a chick), which is entirely unrealistic. In reality, deaths will occur through-out the year. Had the formulation been “At the end of the year […]” this would have been OK—unrealistic, but without ambiguity. However, this is not the formulation used. Now, the formulation used is inconsistent and ambiguous, and the “at” interpretation is a quite reasonable way to resolve the issue—but it is not the only way: The resolution could equally be “[…] will have died.”, which is consistent with a more realistic model and is what would be expected, were we dealing with a real-life penguin situation. Unfortunately, with this resolution the problem becomes under-determined…

    *The less bright tend not to see such complications, which can be to their advantage when it comes to simplistic tests—but to their disadvantage (and science’s…) when they try to become scientists.

    **As was I, but I had the leisure of not being under time pressure; and have enough knowledge of poor test questions to come to the “right” conclusion fairly fast.

  6. “SAILING SHIPS” deals with a technology that seems dubious and/or where weird fictional data have been used to describe a real technology. The inclusion of an apparently actual trademark (“by skysails”) makes it outright shady—is this a commercial plug?

    Notably, the intention is to use a sail attached to a ship by a line, hovering considerably higher up than a regular sail, because “the wind speed is approximately 25% higher than down on the deck of the ship”. Now, this would probably imply a maximum of 1.25 * 1.25 = 1.5625 gain in “push” (both the number of air molecules hitting the sail and the average momentum of individual molecules increases by a factor of 1.25), but with a minimum that could be considerably lower, because the faster the ship goes the lesser the net air speed and the lesser the advantage. At the same time, one example seems to aim for a 45 degree angle, which would divide the force into components, with a proportion of 1/sqrt(2) going horizontally and the same (uselessly) vertically. We then have a maximum gain of 1.5625/sqrt(2) ~ 1.1: The 25% higher wind speed has resulted in a 10% improvement… Barring other advantages (e.g. the possibility to use greater sails) this is hardly worth the trouble. True, the 25% higher wind speed could still give a higher overall speed by more than 10%, because the positive force will only cease after the ship hits the wind speed; however, firstly a higher ship speed means a greater loss in terms of water and air resistance, secondly this technology is not intended for pure sailing ships, but as a help for diesel ships. If the data provided are realistic, I am puzzled as to what the actual point would be.

    Or take specifically question 4: A sail is here alleged to cost 2 500 000 zeds*, while diesel costs 0.42 zeds per liter, which implies (with some other assumptions made in the text) that the sail will pay for it self after 8.5 years! Compare this to the reasonably to be expected costs for regular sails and consider the risk that the sail has failed and needed replacement or extensive repairs before 8.5 years. Sigh… An online source gives the current price of diesel as “1-Dec-2017: The average price of diesel around the world is 0.99 U.S. Dollar per liter.”, from which we can give a rough Dollar estimate of the sail price as 5.9 million**—what the fuck?!?!

    *A fictional currency used for several examples.

    **2 500 000 zed * 0.99 dollars/liter / (0.42 zed/liter)

    (If I were to analyze the technology more thoroughly, as opposed to a test dealing with the technology, I would have additional objections and/or points needing clarification. How, e.g. is the sail handled during a storm without having to cut it loose, to a horrifying loss of money?)

I probably had more objections when going through the questions the first time around (with the purpose of solving the problems), but I have lost my energy here, being about half-way through on my second iteration (with the purpose of writing this post). There was definitely at least one case of “faster speed” or something in the same family, showing a conceptual confusion that no mathematician should underlie: A vehicle can be fast or slow, but its speed cannot; an item for sale can be cheap or expensive, but its price cannot; etc.

As a final note: There was a third question that I failed, namely “Question 2: REVOLVING DOOR” (i.e. the penultimate question). Lacking in concentration, I calculated (I hope, correctly) the linear width of the opening, but the question actually asked for the “arc length”. I take some comfort in the arc length being easier to calculate, but would of course still, correctly, have been marked down.

Written by michaeleriksson

December 17, 2017 at 12:40 am