Some thoughts on generalization

The area of generalization, extrapolation, abstraction, analogies, etc., can be quite interesting—as can the question of how to best handle it. (For simplicity, I will mostly speak in terms of just “generalization”, and the examples might be tilted towards specifically generalization, but the word should be taken in a widened meaning of “generalization, extrapolation, […]”.)

For instance, with most* texts that I write, I find that the contents apply (in whole or in part; with or without relevant modifications; with or without abstraction; whatnot) to other areas. When should I mention this in the respective text resp. when would the mention bring value to the reader? This is a judgment call, which usually turns out as “don’t bother” with me. Factors to ideally consider include how obvious the generalization is (mentioning the too obvious can be a waste of the reader’s time and/or seem insulting), how far-going a generalization might give a benefit, how many possible generalizations there are (the more there are, the more work there is likely to be), the issue of how much text is needed to make the generalization worthwhile,** etc. Then there is the observation that spending more time thinking about a topic might bring forth a slew of new generalizations, which could lead to never-ending work.

*And chances are that the exceptions arise from my not having spotted the generalization(s) yet—not from an absence of possible generalizations.

**I might just mention that “X generalizes to Y”, but this need not be that interesting without a deeper discussion of the details and consequences. (This, especially, in areas like math. An advantage of usually writing about politics and similar topics, not math, is that the additional discussion needed can be much smaller.)

In most cases, however, I do not so much engage in a deep analysis as rely on the mood of the moment—if I at all remember that there was a generalization that could be mentioned. A recurring factor in my decisions is that I often am a little tired and/or tired-of-the-topic when I have a text done, and adding another paragraph to discuss generalizations is not very enticing at this stage. Here we also see an example of how trivial or obvious the generalizations can be. For instance, the same idea applies to virtually any tiring activity, or situation of being tired, that has an optional continuation—but that goes without saying for most readers. And where should the generalization be stopped? Replace “tiring” with “boring” and something similar applies. Ditto “painful”. Ditto even normally positive things, once we enter “too much of a good thing” territory. Abstracting and generalizing to find some single formulation might bring about the tautological “when I am disinclined too continue, I am disinclined too continue”, which truly is too trivial to bother with, might be too detached from the original situation (as being “tired” does not automatically imply “disinclined too continue”), and might still not be the end of the scale. (For instance, a similar idea might apply to a great many other contexts; for instance, a “when X, then X” is a further generalization, just of a different type.)

Similarly, that I might “rely on the mood of the moment” over a deep analysis is not unique to this situation. It can also affect e.g. what I buy in a grocery store, and the generalization starts again. But now we have two different ideas that both generalize, which allows us a generalization about generalizations…

An interesting complication, in a generalization (!) of an older text ([1]), is that adding a generalization to some ideas could conceivably raise an expectation in the reader that I add generalizations whenever I am aware of one. If there is a too obvious generalization of another idea, or a second of the first idea, that I do not mention, then I might look foolish in front of this reader. Of course, the fact that I occasionally have such concerns, while the typical reader is unlikely to even care or notice, generalizes another portion of [1]. Potential further generalizations of this generalization include that “many pay too much attention to the opinions of others”, “many overestimate how much others might care”, and “many fear non-existent threats”, with further generalizations of these possible. Then we have the conflict between my intellectually knowing that few readers will care/notice and my instinctually imagining that they will, usually followed by a quick suppression of my instinct by my brain—which, you guessed it, generalizes. (I will not mention further cases in the continuation, but they are plentiful.)

Of course, the amount and direction of generalization that is appropriate in a given context need not be the same in another context. For instance, if someone working on a specific physics problem makes a novel mathematical observation, this observation is likely to have an analogue in other problems and other areas, where the equivalent math appears, but this might simply not be of immediate interest. For someone working on such another problem, the situation might be different, but it is not a given that more than a one-off generalization to that single other problem is wanted. However, once a mathematician with the right interests gets his hand on the original observation, it might be generalized one or two steps fairly rapidly—and another one or two steps when some mathematicians with other, especially more abstract, interest gets involved. Etc.

It might even be argued that the ability to find the right level of generalization for the task at hand is more important than the ability to find generalizations. (And this level might in many contexts be “no generalization”.)

However, generalization is often something positive, for instance as a means too avoid reinventing the wheel, which can all too easily happen when workers in different fields encounter similar problems. Consider e.g. how often different physical phenomena are, at least too a decent approximation, governed by the same differential equations and how wasteful it would be to develop the same methods of solution in the case of each individual phenomenon—possibly, including the repeated development of the idea of differential equations… Mathematicians are particularly keen on such generalization, e.g. by showing that a certain set and associated operators match a known “algebraic structure”, after which they know that all results of that algebraic structure applies equally to the new case.

In other cases, a failure to abstract can be outright wasteful or harmful in other ways. Consider various arts, including painting and the theatre, where there has been a long history of new artists trying to outdo the previous generation in e.g. the breaking of norms, the “shock value”/provocation, and where to draw the border between art and non-art.* But why? If someone manages to find/create something truly thought-worthy, truly original, truly unforeseen, truly value-bringing as an extrapolation, whatnot—by all means. This has rarely been the case, however: most of what has been considered provocative has been well within what even the layman has been able to imagine on his own, has been an natural extrapolation of previous provocation,** has long been exceeded in less “artsy” contexts,*** or similar. I have, e.g., encountered fictional depictions of artists that have gone as far as to consider murder an art, incorporate murder in performance art, murder for artistic provocation, use body parts of a murder victim as art, and similar.**** What could a real-life artist do that would move me beyond the borders of what I have already seen in fiction or could myself conceive? Splash a bucket of pig’s blood on an empty canvas and call it art? Please! Why not just draw the natural conclusion that this type of provocation, escalation of provocation, whatnot is pointless and will often do more harm than good to the art at hand.*****

*As opposed to e.g. just experimenting further in some direction for more artistic purposes, say, to find out what the effect on a canvas is when a certain school of painting is pushed further and whether the result is worthwhile.

**In principle, if not in detail. For instance, if we start in a very straight-laced era of movie-making and have a “shocker” of showing a young woman in a still-covers-a-lot bikini, the next escalation of shock might consist of a showing a young woman in a modern bikini, showing a topless young woman from behind, or similar. To predict the exact escalation is hard; to predict the general nature of the escalation (and the risk of an escalation) is a different matter. Even an escalation to, say, a completely naked young woman having sex on the beach would be more a matter of quantity than quality, of taking several steps of escalation at once. Going in another direction, imagining a young man or an old woman in a bikini does not take a revelation either—but why would anyone wish to see them?

***Contrast e.g. a sexually explicit art movie with a porn movie.

****Note e.g. portions of “Dexter”, but the idea is somewhat common.

*****Unless the artist follows some disputable non-artistic agenda, e.g. to change societal norms, to ruin this or that art form for the “wrong” persons, or, even, to ruin art. While I do not think that such an attitude is common, it is certainly possible, compatible with the behavior of many political activists in other areas, and compatible with some other excesses. Consider e.g. how some seem to take the laudable attitude of “function should take priority over aesthetics” and amend it with a despicable, unreasonable, and irrational “ergo, we should deliberately strive to make buildings ugly”.

More generally, it is often the case that certain ultimate extrapolations and generalizations follow immediately to the reasonably intelligent, but that mid- or nitwits, who are themselves poor at generalization, try to take each individual step at a time. A good (fictional) example is found in Lewis Carroll’s “What the Tortoise Said to Achilles”, where just one or two iterations should have been enough to prove the point, but Achilles was too dense to understand this—and, maybe, the turtle too dense to understand that he had just caught himself in a trap, where his best bet would be to wait for Achilles to fall asleep and then make a crawl for it. An interesting parallel to this is the idea of plus-one, infinity, and infinity-plus-one:

Consider two kids arguing over the size of something, say, who has the greatest desire for a last piece of cake. A stereotypical dialogue might then include something along the lines of “I want it more than you!”, “Hah! I want it twice as much you!”, “Hah! I want it thrice as much you!”, etc., until someone drops the bomb of “I want it infinity times as much as you!”. Exactly how to consider the introduction of infinity is a tricky question, but something like “extrapolation of what events are likely, followed by an attempt to defeat the extrapolation” might be close to the mark, which could be seen as a case of successful generalization (in the extremely wide sense used in this text). Moreover, infinity as such could* be seen as an extremely interesting extrapolation of large numbers. However, we also see a failed generalization, as both kids, unless first-time participants, should have realized that infinity was coming and that whoever first dropped the bomb would “win”.

*The conditionality hinges on whether it is an extrapolation, not on whether it is interesting.

In a next step, we could have the other kid either conceding or trying some variation of “infinity plus one”—and their argument might then have turned to whether this “infinity plus one” was or was not larger than infinity. Here we land at a very interesting question, as mathematicians consider infinity-plus-one in the sense of addition equal to infinity, meaning that attempting to trump infinity with infinity-plus-one is as pointless as trying to trump three with three (resp., above, thrice with thrice). In this sense, the argument could finally degenerate into whether an “infinity vs. infinity” standoff should be considered a draw or a victory for the first invoker of infinity. But, while mathematicians consider infinity + 1, infinity + 2, and even infinity x 25 equal to infinity, they also have a generalized version of the “successor operator” implied by plus-one.* Here we have a generalization arguably bringing something more interesting than even infinity—the idea of a number larger than infinity.** In the unlikely event that infinity-plus-one was intended as this successor operator, not as a mere addition of one, the other kid would have kept the game alive.

*As in 1 + 1 = succ(1), 2 + 1 = succ(2) = succ(succ(1)), etc., for a suitable operator succ, which for integers is simply that same thing as adding one—but where succ(infinity) is something different. By such generalization of the successor operator, we then have a hierarchy of non-equivalent infinities. The “vanilla” infinity presumably intended by the triumphant kid would then carry the more specific name aleph0. (Assuming the most common approach of using “cardinal numbers” and with some oversimplification, as we potentially begin to compare apples and oranges.)

**But this idea might have originated from another line of reasoning, e.g. Cantor’s famous “diagonal proof” that there are more real than rational numbers, and its generalizations.

However, from here on, it is trivial to infer the possibility* of applying a successor operator to infinity infinitely often to form a super-infinity, a version of the successor operator that finds an even larger successor to the super-infinity, a super-duper-infinity and a successor to the super-duper-infinity, etc. Even trying to break out of this by e.g. constructing a super-duper-whatnot-infinity where the “whatnot” incorporates an infinity of terms stands the risk* of failing due to an even more generalized successor operator.

*As case has it, all these successors, super-duper-infinities, and successor operators exist, but their existence is not a given from the above. Without further investigations, we cannot infer more than the possibility. (This with some reservations for what qualifies as “existing” and what has what semantics.)

Disclaimer:
Apart from some minor editing, the above was written some weeks ago. At the point where the text ends, I was distracted by a text from my backlog, with more mathematical content, which fit well in context and would have clarified a few points above. Having written most of it, I found some issues that I wanted to mull over before finalization and failed to get around to it, which has led to these “some weeks” of delay. To avoid further delays of the current text, I have decided to put the other text back in the backlog. (Especially, as I could benefit from improving my markup language with regard to math before proceeding.) It is possible that some additional thoughts or sub-topics that I intended to include in the current text have been forgotten during this delay. Certainly, trying to go easy on the mathematically unknowledgeable, I run the risk of being sufficiently approximative with the truth as to annoy the mathematically knowledgeable, while not giving enough details for the unknowledgeable to be truly helped.

To, however, give two core ideas of the other text: (a) When we generalize a certain type of number, a certain algebraic structure, whatnot, there is rarely or never one single generalization, and statements made under the assumption of a single generalization can be faulty or simplistic. (E.g the claim that the square-root of -1 is i and/or -i, which truly amounts to something like “the field of complex numbers has the field of reals as a subfield and the number i from the field of complex numbers has the property that i^2 = -1 and (-i)^2 = -1”, which does not automatically preclude that some other field or other algebraic structure than the complex numbers has similar properties and provides another set of “roots”.) (b) The discussion of whether e.g. complex numbers and various infinities exist is in so far pointless as we can just abstractly define some set of elements and operations on these elements, use them when and where they happen to be useful, and forget questions like whether e.g. i is something real (non-mathematical sense) and/or something that “belongs” with the real (mathematical sense) numbers. For instance, the field of complex numbers can be quite useful in dealing with, say, calculations on electricity and magnetism, regardless of what nature we consider i to have—and there are fields equivalent to the complex numbers that do not even mention i.

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More on medians and bids / Follow-up: Oddly equal elections and game theory (and some other thoughts)

My recent in-the-middle-of-the-night attempts to explain math ([1]) have left me with a bad conscience. I will give myself a do-over with a clearer head and try to be a little more pedagogical and to aim at a reader with little math background.* (Text in footnotes and excursions might be more demanding, however.) I will also fill in a few things left out from the original text. As a remark on notation, I follow the common convention of using expressions like “[1, 2]” to denote (real) intervals, here 1–2, with endpoints included; this is not to be confused with “[1]”, which is a reference to another work. I will give series in the form “1, 2, 3”; while all my examples use positive integer entries in the series, this is just a convenience and e.g. “-1.2, 0, 3.14, 9.9999999” would be perfectly possible (also note an excursion with non-integer use). I try to keep bids (in the sense of [1]) in quotes, but might have missed some cases.

*Caveat lector: Pedagogical by my standards. For instance, proof-reading this time around, I spotted quite a few uses of “element[s]”, which seemed the natural and generic choice to me—but I have no clue on what proportion of readers will understand them. (I replaced most of them with “number[s]” or “entr[y/ies]”, which, while more restrictive, are less likely to cause confusion.)

Firstly, what is a median? Broadly speaking, the middle entry of some numbers in terms of value.

Consider the series 1, 2, 3. In this case, 2 is in the middle in the sense that an equal number of other entries are larger resp. smaller. Ditto the series, 3, 1, 2, as it is value, not order, that matters; and I will silently assume that various series of values have been sorted, e.g. in that an original 3, 1, 2, has been replaced with 1, 2, 3. Now, with a sorted series,* we can just remove one number each from the left and the right side, until we have one or two numbers left. If it is one number, that number is the median; if it is two, the median is conventionally defined to be the (arithmetic) average of the two. For instance, applying this to the series 1, 5, 6, 9, 11 leaves us with 6, which is also the median; for instance, the series 1, 5, 9, 11 leaves us with 5 and 9, and a median of 7 (i.e. (5 + 9) / 2).

*For my current purposes, a finite number of entries can be assumed. In other contexts, infinite series and sets with even larger cardinalities can occur. Whether it makes sense to define a median in such cases is disputable, e.g. because the interval [0, 1] has the same cardinality as the interval [0, 100000000], but by using a more generic “equal or larger than” / “equal or smaller than” reasoning some approximate equivalent can often be found, e.g. by putting the median or “median” of the interval [0, 1] at 0.5.

A complication is that there are series (e.g. 1, 2, 2, 2, 99, 99) where values occur repeatedly. This is not harmful, and we can choose an arbitrary sorting of equal entries. Once we have a sorted series, we can proceed as above. Throwing away entries from the ends, we now land at two remaining entries, 2 and 2, with an average of 2. (The alternate series 1, 2, 2, 2, 99, 99, 99 would give one entry, 2 for the same median, but 1, 2, 2, 99, 99, 99 would give 2 and 99 with an average of 50.5 as the median.)

Secondly, why is the median so good a choice in bidding contests like those used in [1]?

As can be seen, the medians fall into two categories: those where the median is actually present in the series and those where it is not. (Contrast 2 and 50.5 above.)

If the median is present, then the original idea that the median wins is virtually obvious: If the value is also unique (as with e.g. the series 1, 2, 3) the number of entries must be odd, i.e. of the form 2n + 1, as can be seen by the above method for finding the median (with n corresponding to the number of rounds of removals).* Bidding the median now gives at least n + 1 entries, regardless of whether the competing bid is larger or smaller than the median, while the competitor will get at most n entries. For instance, in the series 1, 2, 3, 4, 5 even bidding 2.99999 will only give two entries, 1 and 2, to the competitor, while the median, 3, takes 3, 4, and 5 for three entries. 3.00001 fails in a manner similar to 2.99999. If the median is not unique, it will do even better than if it had been unique, as e.g. 1, 2, 3, 3, 4, 5 and 1, 2, 3, 3, 3, 4, 5 have more entries guaranteed to go to the median, namely those that equal the median in value.** They can be considered “improved” versions of 1, 2, 3, 4, 5.*** (Also see excursion.)

*More strictly, every time we end with one entry after removals, the number of entries must be odd. A unique value, however, can only be the median by being the one last entry remaining. (A median of, e.g., 2 can arise from a single entry of 2 or as the average of two entries. If 2 is a unique entry, however, it cannot be the average of two entries both equal to 2. The average of e.g. 1 and 3 is also 2, but this cannot be relevant when 2, as assumed, is present in the series, as 1 and 3 could not be simultaneously among the two entries to average.) Note that a non-unique median does not necessarily correspond to an odd number of entries. Consider 1, 2, 2, 3.

**Note that this holds regardless of what side of the median they are on. Consider e.g. 3, 3, 3, 3, 4, 5 and 1, 2, 3, 3, 3, 3 where the median is guaranteed to gather at least four entries, and any competitor will reach at most two entries.

***But note that the entries in the previous footnote are “improved” versions of 3, 3, 3, 4, 5 resp. 1, 2, 3, 3, 3—not 3, 4, 5 resp. 1, 2, 3. Just removing all duplicates of the median can lead to a series with another median, which breaks the approach.

If the median is not present in the series, however, we have the loophole mentioned in [1]. Take the series 1, 2, 4, 5 and apply the above way to find the median. We find the two entries 2 and 4, with an average of 3, which is the median. However, because 3 is not actually present in the series, there are other values that can reach a draw. For instance, the bids “2.1” and “3.45” would gain the same number of entries (two) as the median in a competition. Generally, any value in the interval [2, 4] draws with the median.*/** This is a reverse of the above, in that there are no numbers guaranteed to go to the median.

*This is ultimately because the definition of median, in the “two entries left” case, is arbitrary. Taking the average is convenient, natural, and in the spirit of the median, but there is nothing magical about the average (a point halfway between the two entries) compared to, say, a point one-third of the way between the entries.

**This leads me to the “medianoid” mentioned in [1]. Create an equivalence class of all the numbers in the interval [2, 4] (or whatever might apply; note that the interval reduces to an exact number when the median is an entry) and consider them the same for the purposes of the competition. Replace “median” with “medianoid” in the recommendations and the loophole disappears. (In a “perfect knowledge” game the equivalence is perfect, in that the decision to bid e.g. 2.5 over 2.6, or vice versa, is wholly arbitrary. However, in a game involving some amount of guessing, there might be a priori reasons to choose the one over the other, while the a posteriori equivalence is more coincidental from the point of view of the bidder. Note that the chance of bidding a “medianoid” value by chance can be considerably larger than that of bidding the median proper.)

Thirdly, I had not done the math on whether there might a multi-issue bidding contest with no unbeatable bid. (By analogy, consider the principle behind rock–paper–scissors.) Had I spent a little more thought, I would not have needed to do the math, as there obviously are. Consider even a single issue* where there is an open ended scale and sufficiently many voters have a preference for “more” that it is not satiable.** The first party bids “100”, the second “200”, the first “300”, etc. No matter how large the bid is, an even larger bid will beat it.

*Here there is a flaw in [1], as I only considered the more common scenario of some finite median and satiable wishes, where deviating, in either direction, from that median is harmful. Above, however, no median exists. (Unless we argue that the median is infinite, in which case the median cannot be reached by a finite bid, with the same result.) The scenarios chosen in [1] are a little restrictive in other regards too. (Consider e.g. the discrete “0”, “1”, “2” bids below.)

**No, this need not be unrealistic even with adult voters and a national election, although it might require an extraordinarily positive issue (and although I can give no example off the top of my head) or the voters might need to be unusually egocentric and/or irrational (to e.g. see more “free” money as good, no matter how much more and with no thought on mid- and long-term side-effects). If in doubt, however, we can consider a vote among children who are promised certain quantities of candy or whatever they might prefer, or a vote in a sufficiently small circle, e.g. where the few voters are to be given money from others and these others lack a vote.

A more interesting question might be whether there is some more finite combination, say on scales limited to a certain interval. Again, the answer is affirmative—and again so simple that it works even on a single measure. There might simply be a rock–paper–scissors situation, that the voters prefer a bid of “1” to a bid of “0”, a bid of “2” to a bid “1”, and a bid of “0” to a bid of “2”. (Where no other bids than “0”, “1”, “2” are possible.) Interestingly, this can be a quite realistic situation, e.g. when the numbers represent specific politicians and the voters have this circular preference chain for the three* politicians put forward (but two at a time, one for each of the two bidders/parties). We might even have a scenario where voter preferences rise in tandem with a continuous bid, but only up to a critical point, at which 0 is suddenly more attractive and things start over.

*To allow the parties to field different candidates, it might be better to have four of them, with obvious modifications to the 0–1–2 chain.

Fourthly, I had not done the math on whether (a) a multi-issue bidding context might fail to have a median-like solution and (b) whether there might be more than one constellation of bids that were equally good.*

*Even the “medianoids” aside; and apart from rock–paper–scissors situations. A case can be made that the latter form three (or more) equally good bids, as none is superior or inferior to the others overall. However, my intention was on bids that were equally strong at any given moment, which is not the case with rock–paper–scissors. Imagine an alternate reality where rock and scissors both beat paper and draw against each other.

Concerning (a), it comes quite close to the “Thirdly” above, and it is also touched by (b), and I will leave it at that.

Concerning (b), I will first give a case of two two-issue bids, “1”/“0” and “0”/“1”, that have the same result in a certain situation. Say that the voters’ reactions are governed by a preference function f(x, y) = x + y. Clearly, f(1, 0) = 1 + 0 = 0 + 1 = f(0, 1), implying that “1”/“0” and “0”/“1” are equally good bids. This was not quite what I had originally intended, however, which was multiple bids that tied for being the best bid—and clearly better bids, e.g. “1”/“1”, exist for this function f. Consider, instead, f(x, y) = (x – y)^2, with the restriction that x and y are both >= 0, <= 1.* Now, f(1, 0) = 1^2 = (-1)^2 = f(0, 1), and no higher value can be found, as we effectively see a maximum when x and y are as far apart as possible.

*Similar restrictions are not that unusual, if possibly after some normalization. Consider e.g. income tax: a negative tax rate or a tax rate exceeding 100 percent is theoretically possible, but would be a sufficiently rare suggestion that a tax rate between 0 percent and 100 percent can be virtually assumed, if in doubt because a realistic preference function would find far too few voters willing to accept tax rates outside that interval. (Or consider a percentage for student-loan forgiveness or for how much of the capacity of a power-plant is to be used.) In contrast, an inflation rate outside that interval is very possible and has happened on many occasions throughout history.

Excursion on bids like “2.1” and “3.45” vs. simplifying assumptions:
In [1], I made the simplifying assumption that all bids are integers. This does not close the loophole. Firstly, look at the context where the bids “2.1” and “3.45” appeared above and note that “2” and “4” have the same effect. Secondly, a “medianoid” can contain other integers than the median (should the median be an integer) and the endpoints (should they be integers). For a trivial example, consider the series 0.1, 10.2, with median 5.15 and “medianoid” [0.1, 10.2], where there are ten integers within the “medianoid”, but where neither the endpoints nor the median proper is an integer.

Excursion on medians and “improved” versions:
Given a (sorted) series with a median, we can describe it by five non-negative integers: m0 for the number of entries at the exact point of the median (1 or 0; can be seen as the number of remaining entries at the end of the removal algorithm modulo 2); m+ for the number of entries that match the median in value, but are at higher positions in the series; m- as the same but at lower positions in the series; o+ as the number of other entries higher in the series than the median (or equivalently the number of entries larger than the median in value); m- as the same but lower in the series (or smaller in value).

(Consider 1, 3, 3, 3, 3, 3, 3, 4, 5 with median 3. Here m0 = 1, m+ = 2, m- = 3, o+ = 2, and o- = 1.)

Note that, for m0 = 1, the total number of entries is 2n + 1, for some n, and m+ + o+ = m- + o- = n; while, for m0 = 0, the total number of entries is 2n, with the same equalities holding.

A bid equalling the median is guaranteed to have at least m0 + m+ + m- entries. A competing alternative bid can do no better than max(o+, o-); and the median then receives another min(o+, o-) entries. Assume that o+ results from the optimal alternate bid. (The following, with obvious modifications, will hold equally for o-.) This now leaves us with o+ entries for the competitor and m0 + m+ + m- + o- entries for the median. As m- + o- = n, this equals m0 + m+ + n; while, conversely, o+ = n – m+ <= n and o+ = n – m+ = m- + o- – m+ = o- + (m- – m+)*. As we can see, if m0 = 1 or if m+ > 0 the median is guaranteed to win outright, while a tie** results exactly when both are 0.*** Moreover, if m0 = 1, m+ and m- can be pairwise increased or decreased (provided that neither turns negative) without altering the victory, as these pairwise changes keep the same median.**** For m0 = 0 and assuming that the median is an entry in the series, both m+ and m- must be >= 1 (as at least one entry on both sides is needed to form the average), and we have the same claim about pairwise increases/decreases as long as both remain >= 1.***** This explains claims like 1, 2, 3, 3, 4, 5 and 1, 2, 3, 3, 3, 4, 5 being “improved” versions of 1, 2, 3, 4, 5—the victor is guaranteed to be the same for series that differ only in pairwise increases/decreases of m+ and m- resp. a sufficiently “kind” change of m0—but the larger m0, m+, m- are, the better the “score”.

*This might seem paradoxical at first view, as o- and o+ seem like they can be freely chosen. However, if they are not increased in keeping with the last equation, the median changes, either in that an entirely different value ends up being the median, or in that the position of the median within the range of entries that had the same value changes, which causes a corresponding change to the values of m+, m-, and (maybe) m0.

**Note that a sub-optimal bid from the competitor can still lose in this situation.

***This is a more algebraic proof of the “median wins” idea above, but this is not the point of the current discussion.

****In contrast, non-pairwise change could alter the median. Cf. the above footnote on o- and o+; and see a much earlier footnote for an example. Note that while the victor does not change, the margin of victory will do so.

*****If both reach 0, the median is no longer an entry (and will likely fail to be the median), which opens the door for the loophole. If the one reaches 0 and the other does not, then the former median will cease to be so, and the new median will be the average of the old median and whatever number it is paired with after applying the removal algorithm. However, in both cases, the (old) median should be a member of the (new) “medianoid” and guaranteed at least a tie.

A few thoughts on educationrealist

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?”, …

A critical look at PISA

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.