This text was originally written in April 2023, during the Dark Ages of Wordpress. For some reason, it appears that I failed to actually publish it, despite it being done or nearly done. I discovered this in December 2023 and published it on my website proper instead.
This did entail some changes due to differences in the markup languages used, notably a switch from footnotes to side-notes, and could conceivably have left some missed issue in the text. A side-effect of the switch to side-notes is that the fit of note to main text is not always optimal, as I have mostly just changed the markup and not the marked-up text; another that there are more side-notes than even I would normally write. Likewise, there might be some issues arising from differences in approach between blogging and regular “websiting”.
(A below claim about poor math capabilities of the markup language holds in both cases, at least at the time of writing.)
Some textual changes have been made, including an excursion on “continuous evaluation” and a move to speak of just “show[ing]”, to reduce repetition of “show[ing] the work”. (Generally, texts published on this site are not intended to be static and many have changed over time.)
There is a common practice of “showing the work” in math, e.g. in that a high-school math student is supposed to show how he arrived at a particular answer to a particular problem, instead of just giving the answer. In the last, maybe, six months, I have repeatedly seen demands that “showing the work” be abolished, because it would be “White Supremacy”, “discrimination”, or similar—or, in other words, not all students do equally well, and “showing the work” reveals the differences to the disadvantage of those who do less well. This is particularly interesting in that it was usually, in my own school experiences, the brighter students who had an aversion to this (I certainly did) in high school.
Terms like “brighter” and “duller” are relative and must be seen in comparison to both the other students at hand and the problems at hand. Moreover, there is often a factor of being better or worse prepared, which I, for the sake of simplicity, will largely gloss over. (This especially as the brighter students will need less time to understand the study materials and likely typically need less preparation to reach a certain effect.) The same “principle of relativity” might apply elsewhere, e.g. for where to draw the border between head-work and leg-work for any given student, which can have side-effects like a brighter student entering leg-work territory earlier than duller students. Moreover, many statements should be seen with an implicit “on average”—it is, e.g., possible that a bright-but-lazy student does worse than one dull-but-industrious.
First, consider why many (but by no means all) brighter students can be reluctant to “show”: Assume that you are one of these students, and that you are now faced with a problem with an obvious-to-you solution (see excursion for potential examples). You know exactly how to get the answer, often you can do the calculation in your head, and, if not, making one or two brief bridge calculations on paper or just jotting down one or two numbers as a memory aid might be all that is needed. Getting the answer might involve seconds of work—but “showing” can involve an effort/a time that is many times larger. You could now “show”—or just write down the answer, proceed to the next problem, and be done with the test in no time. (The more conscientious student might use the spare time to triple-check the answers and leave a bit early; the less conscientious might do less checking and leave the more early to, e.g., do some reading.) To this, also bear in mind that brighter students tend to prefer head-work to leg-work, the former often being interesting and stimulating, the latter boring and demotivating.
In some cases, although more in homework situations than on tests, the difference can be enormous, e.g. if the problem/exercise/whatnot includes a lengthy and repetitive low-value task. See excursion.
Contrast this with a duller student and the same problem (or the original student and a much harder problem), who might have to spend minutes thinking of/on a solution, take several approaches until one sticks, spell out all the calculations on paper in order to be sure that they are correct, whatnot. Firstly, the relative effort of “showing” will be much smaller. Secondly, the “showing” part might be absolutely smaller, as the work already exists on paper, and now only has to be transcribed in a manner legible-to-others.
In my specific case, my poor penmanship was an additional deterrent, as writing in a legible-to-others manner cost me unusually much time.
Here, we also see the issue that a further dumbing-down of school might be outright harmful to the grades, test results, whatnot, of the brighter students. (As well as, obviously, to the learning outcomes, and not necessarily restricted to the even semi-bright. This is more important in the big picture, but also something that most understand to begin with.) On the one hand, they will see a smaller potential improvement through a greater mastery of the dumbed-down material than the duller students, because they already had great degree of mastery and the distance in results will shrink (and, no, there are no bonus points for finishing a test early). On the other, they will stand a greater risk of being bored, feeling that “showing” and whatnot is pointless, or otherwise be moved to put in a lesser effort. (Similar factors apply when homework with more “leg” than “brain” and/or poorly implemented “continuous evaluation” schemes take the place of tests or, more generally, checks for mere effort take the place of checks for actual accomplishment/mastery/whatnot.) The secondary results can then include grades that are not only too lacking in granularity due to grade inflation, but which can also be outright misleading, as e.g. a brighter-but-demotivated student can receive a worse grade than a duller-but-does-all-the-homework student—even when the former understands and is able to apply the material better than the latter. A potential other negative side-effect, and one that I suffered from myself, is that the bright student picks up poor habits that become a hindrance beyond some point of uni, when the difficulty of the courses catches up with the brightness.
I was tempted to write in terms of e.g. “duller-but-does-all-HER-homework” above, as boys seem relatively more likely to be in the brighter-but-demotivated group and girls in the duller-but-does-all-the-homework group. However, while this makes the problem worse and shows how destructive modern educational thought can be, there are plenty of exceptions on both sides.
Second, consider why “showing” can be important: Giving the right answer does not automatically imply that one understands how to arrive at the answer, and it cannot be ruled out that the student made a lucky guess, arrived at the right answer for the wrong reason, or, even, cheated—and a cheater is more likely to find the right answer through cheating than the actual road to the answer. A good example from my own high-school days involved a somewhat complicated expression adding and subtracting the numbers 4 and -4, with repeated absolute values inserted. Naturally, I do not remember the exact problem, but consider something like ||4| + |-4| - 4| - |-4 + |-4||. Even someone with a poor grasp of absolute values might understand that the answer has to be an integer multiple of 4 and likely a comparatively small one (say, 4, 0, or -4). This student might now wager a guess that the answer is “4” and be correct. Another student might perform the calculation incorrectly—and still arrive at “4”. Here, it is vital to see the “work” in order to judge what points the student deserves.
There can also, for more complicated problems, be an element of giving the wrong answer for a sufficiently good reason that points are still warranted, which naturally hinges on the corrector being able to see the work. That is off-topic, however.
For an example of incorrect calculations giving the right answer, consider the similar problem ((4) + (-4) - 4) - (-4 + (-4)). It has the same solution, while being equivalent to the original problem when incorrectly considering the absolute value the same as the raw value, which likely is a common mistake among mathematically weak high-school students.
An example of the “work” is to replace inner absolute values, simplify, replace, simplify, etc., until a solution has been found. Here e.g. ||4| + |-4| - 4| - |-4 + |-4|| = |4 + 4 - 4| - |-4 + 4| = |4| - |0| = 4. (In contrast, doing the same to the example in the preceding side-note would eventually amount to -4 + 8 = 4.)
Thirdly, consider why some now want to abolish the idea of “showing the work”, of right and wrong answers even in math, whatnot: Well, here I actually have to speculate, as the “argumentation” seems to be based on cheap slogans and racist hate-mongering, but it certainly is not for the reasons of a bright student with an aversion to leg-work. Possibilities include that Blacks do worse than Hispanics, who do worse than Whites, who do worse than Asians, which is intolerable to many Leftist extremists; that members of the strongly anti-intellectual, anti-scientific, anti-whatnot Left honestly do not understand or refuse to acknowledge issues like those discussed under “Second” or how getting the right answer can actually matter in life after school; that any type of ranking is seen as, e.g., a “social injustice” and must be abolished even with dishonest means; a fanatic belief in that absurd and long disproved “nurture only” idea, which gives external factors as the sole explanation for differences in outcomes; and similar.
As noted in an above side-note, the differences in, say, pointless leg-work can be enormous in e.g. homework situations.
Similarly, some extremely poorly written test questions can be ridiculously open ended (and should be grounds for a summary firing). Consider e.g. a “List as many bird species as you can!” (as opposed to e.g. “Give five examples of bird species!”): If the request is taken literally, the highly knowledgeable-about-birds student might find himself bogged down with non-stop writing for minutes, while the rest of the class breezes through—and the more conscientious student might spend a few minutes extra in the knowledge that those few minutes extra might give a few birds more, while the less conscientious might name some few birds and then move on. On the other hand, if taken less literally, the student has no way of knowing when he can safely stop. Indeed, in a somewhat similar situation during my own high-school days, I outright asked the teacher whether I had given sufficiently many examples—and he refused to answer me, because that “would be helping” (or some such). This despite my not having asked whether the examples already given were correct—just whether they were sufficiently many.
Said teacher/idiot was also the test constructor and the later corrector, so no excuses can be found in that area. He was a repeat offender in terms of poor test construction. In another memorable case, there had been a similar question with a given number of maximum points (hypothetically, 7), virtually all other teachers used a “one point per example” approach, I gave seven examples in the belief that this would secure me the full points, but several points were deducted because I had not given enough examples... Well, if the idiot had had the common sense to state how many examples he wanted, or had kept to the same principle of “one point per example”, I could have given enough examples—as is, the key was not to have (very superficial) knowledge about the topic at hand, but to correctly guess how many examples were wanted...
Of course, merely being able to list some certain number of birds or whatnots is hardly a reasonable test for a high-school student. If in doubt, many can do that based on school-independent general knowledge, while something relating to the behavior, anatomy, ecology, whatnot, of birds would be much more valuable. (How typical or atypical these questions were, I do not remember.)
A particularly unfortunate-as-usually-implemented family of problems involve proportionalities and/or powers of proportionalities, through a drive to calculate in detail intermediary values that are simply not needed to solve the problem. Consider a (hypothetical) question like:
You have a solid ball of an unknown material with a radius of 10 cm. The ball weighs 1 kg. How much would a solid ball of the same material and a radius of 20 cm weigh?
Whether “proportionalities” is a good term can be disputed, but nothing better occurs to me at the moment. The intent should be clear from the below, however.
A better problem would focus on mass instead of weight, but this formulation seems more realistic with an eye at my own, if admittedly vague, high-school memories. If in doubt, the failure to make a distinction between mass and weight is a good example of dumbing down and how it can easily do more harm than good to the students. Note that other points of criticism might apply, which matches a realistic school problem, e.g. regarding the silently assumed homogeneity of the material.
Here we see another issue where the brighter students might get into trouble, while the duller plod along: The brighter will spot such inaccuracies and holes much more often and risk losing time filling them, e.g. by stating additional own assumptions or enquiring what additional assumptions are intended.
A bright student is likely to immediately recognize that the answer is 8 kg. If pressed for details, he might add that the weight is proportional to radius to the third power, and that doubling the radius would, then, increase the weight by a factor of 8.
Variations of the same solution are possible for any other relative change, e.g. in that an increase from 10 to 11 would give a factor 1.1 and a cubed factor of 1.331, with a corresponding answer of 1.331 kg. (The coincidental inspiration for the next example.)
While this explanation would likely pass muster as “showing” with all but the most stupid teachers, the typical textbook version of the same solution, and what some might require, seems to be a variation of (excepting the weak math capabilities of my markup language):
The volume of a sphere is given by 4/3 π r^3. The first ball has the volume 4/3 π 1000 cubic centimeter.
The density of an object can be calculated as weight / volume. [“mass” is better, but see above]
The first ball must then have the density of 1 / (4/3 π 1000) = 0.00075 / π kg per cubic centimeter.
The second ball must have the same density. We can now calculate its weight as density times volume.
Using the known formula, the second ball has the volume 4/3 π 8000 cubic centimeter. The weight of the second ball becomes 0.00075 / π * 4/3 π 8000 kg or 8 kg.
Or consider, for a non-proportionality example, something like:
Calculate 14641 / 11.
That the answer is 1331 can be seen fairly easily, e.g. by noting that, if an integer solution exists, both the first and last digit must be 1 and taking it from there—and trivially, if someone is familiar with e.g. Pascal’s Triangle, some basic binomial sequences, or just small powers of 11.
However, just claiming “1331” is unlikely to meet the approval of the corrector. Calculating 1331 * 11 is easy, but might not be allowed as an answer: It is clearly proof that 1331 is the right value, but I suspect that many or most correctors would argue that the missing explanation (for why specifically 1331) is unacceptable. If in doubt, the student might have spotted “1331” on someone else’s test and just performed the multiplication to verify that this was the solution.
No, what is wanted is an explicit division, typically “long division”. Frankly, I do not remember the exact rules and notation (why should I, when I actually understand how division works?), but the result would be more complicated and take longer to bring to paper—and might well be more error prone.
What if the answer is not an integer and/or not as convenient? Well, consider e.g. 14641 / 13, instead. Here, there is no shame in grabbing pen and paper, and I do not argue that this type of division should be done in one’s head (14641 / 11 is an interesting special case). However, 1126 3/13 certainly can be found in the head. (I just did, although I took the caution of verifying the result electronically.) The integer part is clearly four digits long, and the first digit is clearly 1. We next have 1641, and the first of three digits is also clearly 1. Next 341, with the obvious result that 20 * 13 = 260, leaving 81 = 78 + 3.
This is a somewhat controversial topic, there are many who swear by it, and there are many who swear at it.
My own take is that continuous evaluation has little or no place when it comes to setting grades, as it almost invariably shifts the evaluation from a test of mastery to a test of effort. Even in a best case scenario, a test of overall mastery at the end of the course, which is what matters, is (wholly or partially) replaced with tests of partial mastery over time, which can skew the evaluations considerably (even aside from issues like head- vs. leg-work): There are many reasons why someone who has an inferior partial mastery at time A might have a superior overall mastery at time B, including short-term losses due to illness, a higher course load (which might, e.g., require prioritizing the one course over the other in the short term), and a better study technique. Then there is the issue of what is actually remembered at the end of the semester from what was tested during week 1, week 2, etc. By analogy, should we award the victory in a marathon to the runner who has the best time at the finishing line—or to a runner who came in fifth but happened to lead at 10, 20, and 30 kilometers?
In contrast, using continuous evaluation to find out as soon as possible what student might have what weakness, might need what type of help, whatnot, can be very worthwhile. To discover, say, that “John has problems with ‘ie’ vs. ‘ei’ when spelling” after a grade-deciding test gives little room for intervention. However, as far as the teacher is concerned, this benefit is largely limited to school settings. (The student can still benefit even at the college level, and likely more so than during school, as own responsibility is much more important.)
To expand on study technique: Those who follow the lesson plan strictly and/or base their study on lectures/whatnot will typically work some set of pages or chapters at one time, the next set after that, the next set after that, etc. This is strongly compatible with continuous evaluation—but it is a poor approach for reaching a long-term goal. The long-term goal is better met by continuous study, where various portions of the course/class are visited and re-visited repeatedly throughout the course, both to refresh one’s memory and to see the various portions in light of each other.
I have found it particularly fruitful to give an entire textbook a (comparatively shallow) reading at the beginning, so that I have a strong overview of what fits where and have some idea what implications chapter 1 can have on chapter 10 (and vice versa) when a more thorough reading of the individual chapters follows. But try to do this while simultaneously having to study chapter 1 in depth for the first increment of the continuous evaluation—and doing so for several courses that all are continuously evaluated.
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