A first version of 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.)
For the original publication, some textual changes were made, including an excursion on “continuous evaluation” and a move to speak of just “show[ing]”, to reduce repetition of “show[ing] the work”.
Subsequent to the original publication, further changes have been made, if mostly in the form of additional excursions. (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.
An interesting question is whether similar issues might have harmed my handwriting: I objectively was poorly coordinated through much of my school years, but there are at least two things that could have made the problem worse. Firstly, just as “showing” can limit thought with math problems, writing can limit thought more generally. This even for somewhat slower thinkers, as writing with pen-and-paper is a very slow process, and the more so for faster thinkers. Attempts to write faster, even at the cost of legibility, are only natural and the more natural the faster the thinker. Secondly, I was bored to tears, horrifyingly under-stimulated, by the dreary copying of letters that filled much of my early school days—and, likely, more so than most other kids.
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 are likely to see a smaller improvement in e.g. test scores than duller students, because they already would have scored well without dumbing down, 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.
The likelihood that uni courses catch up with the brightness will, of course, also depend on the amount of dumbing down (and factors like field of study). Sadly, the accumulated dumbing down, over decades, has become a massive problem even in tertiary education.
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 the 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.)
From another perspective, the purpose of school is not to show how good one already is—but to become better. (Ditto other forms of training and by no means restricted to the academic.) This central observation is a very strong argument for grading by tests and rare tests at that: In this manner, the exception to the “become better” can be limited to very short phases of “show how good”, with less interference with the process of improvement. Continuous evaluation (if used for grading!) does the opposite, replacing improvement towards a long-term goal with a continual showing of how good one currently is. (Or, worse, a continual showing of how hard one works, how compliant one is, or similar, with little regard for the actual results achieved.)
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.
Yet another problem is that mapping thoughts into language can be tricky, and thoughts into mathematical notation very tricky. For instance, consider 314 / 4 = 78.5. (This problem presented it self to me in real life, shortly before writing this excursion.) My approach was to note that 320 (!) equals 4 * 80, that this overshoots the mark by 6, and that 6 / 4 = 1.5. Correspondingly, we are looking for 80 - 1.5 = 78.5.
Putting the above problem into formal terms and solving it using long division is easy (although the calculation is a bit harder), but how do I explain something like the above using only or almost only mathematical notation? Virtually impossible, as mathematical notation (or, at least, the ones that I am familiar with, let alone what, say, a high-school student or high-school teacher might be familiar with) does not contain true provisions for describing thought processes. To go down the formal road might, then, punish the bright student by forcing him to solve the same problem twice—once in a clever way and once the long-division way, which can be put into notation.
If time allows it, it can be a good idea to solve the same problem using different means, in order to verify the correctness of the answer, to draw new lessons, or similar. However, for such an easy problem this would be boring leg-work and in a test situation it could cost valuable time.
Another complication is that the notation used for long division can vary. For instance, my maternal grandmother learned a different notation in school than my mother did and both a different notation than I did. If we now have the student giving the right calculations in the “wrong” notation, what will happen? (Or, in the context of homework, what if a parent or a grandparent helps using the “wrong” notation?)
At a minimum, we would then need natural language to give a reasonable explanation of the solution used above. My formulations do this job reasonably well for a sufficiently receptive reader; however, writing them down with pen and paper (which was invariably what tests used back in my day) would take much longer than the actual calculation. (While a dull student doing long division might see a single step of writing and solving simply through performing the long division on paper in a sufficiently clear hand. There would be even more work, yes, but there would only be one “batch” of work.)
To continue the division example from the previous excursion, there is no guarantee that an unreceptive teacher/corrector/whatnot would accept that solution. For instance, I have known teachers who would have objected to the claim that “320 (!) equals 4 * 80” with a demand for a proof. However, performing that calculation would incur considerable additional work—the more so if multiplying 4 and 80 was rejected in favor of dividing 320 by 4 to arrive at 80 per long division. (At that point, we might just as well perform long division on 314 to begin with.) The hitch, however, is that a high school student of some mathematical ability would not even need to calculate this, but simply “know” what 320 / 4 is. (Or, on the very outside, near-immediately deduce it from the long memorized fact that 4 * 8 = 32.) This knowledge is what makes the road used by me faster and easier. For instance, some might demand a more explicit calculation to justify the subtraction of 1.5 from 80 to arrive at 78.5. (Also note the earlier example of 4, -4, and absolute values: the leap needed for this subtraction is small enough that it should be obvious to any teacher worth his salt, but (a) not all teachers, let alone correctors, are, (b) there are some students who might actually guess at how to make such simple combinations of numbers, which, if the corrector was not familiar with the student, could give some justification for a request to “show”.)
To the one, then, “320 (!) equals 4 * 80” is known or trivial and needs no further discussion; to the other, it might need calculation. In a next step, we have questions like whose knowledge/understanding/ability/whatnot should be used as a baseline—and, again, the brighter students are at risk, because they form a minority and because the teachers are likely to go by the standards of a majority or plurality that comprises the average and/or the dull. If worst comes to worst, the own abilities of the teacher will be a blocking factor in that a teacher is unlikely to accept, without elaboration, a step that is not obvious to the teacher, himself.
The variation in knowledge/understanding/ability/whatnot, however, can be very large. In the other direction, there might e.g. be someone who actually remembered or immediately “saw” that 314 / 4 = 78.5 in the way that I remembered/saw that 320 / 4 = 80, that 320 - 314 = 6, that 6 / 4 = 1.5, and that 80 - 1.5 = 78.5. (And if someone “sees” that 314 / 4 = 78.5, it is not a given that there is anything to put in words or notation to begin with.)
To make matters worse, the baseline used can vary considerably over time, e.g. depending on how old the students are and what previous mathematical (here, arithmetic) exposure they are supposed to have. Indeed, the type of memorization or internalization demonstrated by knowing that 4 * 80 = 320 is fundamental to math, in general, and arithmetic, in particular. For instance, a long division would likely at some stage involve dividing 31 by 4. (With reservations for my lack of memories of how long division is performed.) Should the student now be allowed to merely posit that this gives 7 and a remainder of 3 as inputs to the further calculations or should he be forced to prove this, e.g. by performing an explicit division of 31 by 4? Chances are that the positing is not only allowed but taken for granted at the high-school level, while something more explicit might be required in sufficiently early years. At the extreme reverse, a first grader might be forced to calculate 4 + 5 by, say, noting that 4 + 5 = 3 + 6 = 2 + 7 = 1 + 8 = 0 + 9 = 9. (While he, later on, might be considered a dunce for not “knowing” the answer without any calculation.) Worse, there appears to be many teachers who teach their grade one students, say, “You cannot subtract a larger number from a smaller number!” (e.g. amounting to 3 - 5), while something very different is taught just a few years later (e.g. that 3 - 5 = -2)—and what about the poor student who has had the misfortune to learn about negative numbers before the teacher considers the knowledge allowed?
Likewise, a more or less advanced math student at the college or graduate level might have very different expectations of the baseline. For instance, in my exchange around grading at Handelshögskolan, my counterpart (concerning another problem) complained that I had correctly found the extreme point for some function but had failed to show that it was a minimum. Here, there is a considerable chance that we simply had someone who (a) was used to comparatively advanced math taking something for granted and (b) being measured by the standards of the average high-school graduate. For instance, if the high-school level problem had been “Find the minimum of x^2 - x + 5!”, it would be obvious to me (and many others) that this minimum is found for the x where the derivative equals 0, because the x^2 is necessarily positive and dominating the expression for sufficiently large and small x, while functions of this form have exactly one minimum/maximum. Indeed, the calculations needed are easy to do in the head, with an x=0.5 and a function value of 4.75; indeed, I would not even need to use derivates to arrive at this, as x^2 - x clearly has “zeroes” at 0 and 1 and symmetry of the function dictates that the minimum is found at 0.5 (and the same 0.5, obviously?, applies to x^2 - x + 5). However, in order to show to the satisfaction of a teacher of beginners, chances are that, among other acts of leg-work, the second derivate has to be explicitly calculated and found to be greater than 0 (it is: 2 > 0).
(2025-09-10)
The above is an indirect and unintentional illustration of how some points can be tricky. Most notably, the first published version spoke of “sufficiently large x”, which reflected a carelessness in formulation—not a lack of insight. (I was thinking in terms of the absolute value of x, but did not write what I meant.) Now, imagine being a corrector given some line of similar reasoning in a solution which speaks just of “sufficiently large x”. (With no mention of “or small”, “absolute value”, or some other modification to the same effect. I speak of “similar reasoning”, because the above would hardly be a good explanation within a solution to a math problem, even if it serves the purpose of illustration above.)
Technically, the claim could be viewed as insufficient, because a key is that the same claim holds true for sufficiently small x (e.g. x=-1000) too, and deducting points would be warranted. (While, in a more advanced context, a critique of a proof for a new theorem should certainly point such a lapse out, regardless of whether the provider of the proof had made an oversight in formulation or in actual thought.)
Then again, in some contexts, it could be argued that the claim is sufficiently complete, because we know that the value at x=0.5 was 4.75 and that this was the sole minimum/maximum, and can conclude that values on both sides of x=0.5 will necessarily deviate in the same direction. (I would be cautious about arguing that case, myself, for the simple reason that the insight and/or reasoning needed to go down that road exceeds the insight and/or reasoning needed to solve the original problem by a fair bit. It is by no means a giant leap, but it is a disproportionate leap. To boot, it is a leap that could trip, say, some high-school math teachers. To boot, can someone, as a corrector, assume that the student at hand could and did perform that leap or must he be suspected of just being lucky? Note again the example with 4s, -4s, and absolute values.)
In other contexts yet, a complaint might be viewed as pedantry, as when two math professors discuss something verbally, and a “You mean ‘large or small’?” might be met with an irritated “Of course! That goes without saying!”. (This the more so, when different speakers have different conventions. For instance, many mathematicians use “greater” to imply “greater or equal” and use “strictly greater” to imply “greater” as used by most laymen, i.e. “>”.)
Other points of criticism in detail can apply, e.g. in that the claim of “exactly one minimum/maximum” refers to a derivative-equals-zero-case and ignores the maxima that occur for x at positive/negative infinity or, for a limited domain, whatever extreme values of x, it self, apply. Depending on details, this might or might not be kosher. It might even be that 0.5 is not part of the domain, which would make for a problem of a very different character.
A potentially interesting point is my above use of “insight and/or reasoning”. While I had no immediate plan with the claim, I suspect, proof-reading, that this juxtaposition captures the overriding problem in a nutshell, in that e.g. prior exposure and prior reasoning leads to insight and that the insight obviates the need to apply reasoning again, but that this can cause problems when the counterpart needs and/or expects reasoning or has a different set of insights and, therefore, sees reasoning as necessary for different aspects of e.g. a math problem. (Also note some similar remarks elsewhere in this text and e.g. how this parallels knowing/remembering the result of 4 + 5 vs. calculating 4 + 5.)
Unfortunately, the exact problem involved was not clear from our exchange. I base the above on another example from from Handelshögskolan that I have in vague memory exactly because thoughts similar to the above paragraph went through my head during my work.
(As the exact problem is not clear, I cannot rule out, unfortunately, that the mistake was mine, that I had been sloppy and forgotten to check for max/min when doing so was actually called for; however, considering the degree of difficulty typically encountered at Handelshögskolan, an explanation like the above is more likely—and the point is to illustrate a bigger issue, not to analyze a specific and actual past situation.)
As for level of students, etc.: Firstly, the math at KTH was far harder than at Handelshögskolan, in general. Secondly, the courses that I took at Handelshögskolan were mostly from the first year, while I had progressed considerably further at KTH (in part, because I began a year earlier; in part, because KTH was my main focus in terms of course load).
Here we also potentially see how a “creative” method can lead to a certain result faster than the “school method”, just like with “my” division vs. long division above. In this specific case, it might have been a tossup, but similar ideas can apply to more complicated optimization (and other!) problems, in that someone who actually understands the problem can find a solution faster and easier than someone who just stubbornly breaks out the formal toolkit. Also see a reverse situation where I was the one who stubbornly broke out the formal toolkit.
But what happens when the teacher lacks the deeper understanding and only has access to the formal toolkit?
Off topic for this page, we also have issues like the student seeing complications that the teacher does not and/or which the student is “not supposed” to see even if the teacher might. (“You can too subtract 5 from 3!!! It’s -2!!!”)
For instance, in the context of elementary optimization and the type of math likely to occur at Handelshögskolan, it can usually be safely assumed that the above x is a real, integer, or similar—possibly, with some range restriction, e.g. x > 0.
However, writing the above, I find myself actively having to suppress disclaimers about complex numbers. Indeed, it is known that an equation like x^2 - x + 5 = 0 has two roots, and clearly, then, values smaller than 4.75 can be found if we include complex numbers. (And note how this is, in some sense, obvious without calculation, but can be hard to demonstrate to someone without the right prior knowledge. As a counterpoint, using complex values for x would usually lead to complex values of x^2 - x + 5, which makes the idea of a maximum and a minimum less relevant; however, real-valued functions of a similar nature can be found.)
In contrast, a business problem might involve the need to adjust for discreteness with an increased likelihood. An x = 0.5 might, e.g., not be an acceptable solution if x is supposed to be an integer, in which case 0 and/or 1 has to be substituted for a result of 5 instead of 4.75. (A trivial claim for someone who understands second-degree polynomial functions but taking quite a bit of work to show to a teacher who does not.)
Now, imagine if I got bogged down in discussions like this when giving a solution to an exam question... (A more advanced analog to the above bird fan who gets bogged down listing birds and might end up scoring lower on the overall test than a less knowledgable student, who has more time to spend on the rest of the exam.)
Since my original writing, I have done some additional reading on topics like “unschooling”. In some source, I encountered a home- or unschooling mother who noted that if the child/student had proved the ability to solve problems like 4 + 5 = ?, she moved on without forcing him to go through an endless list of repetitions of similar problems. (In an approximate restatement. I have forgotten the exact details.)
While the idea is sound, some care must be taken. Certainly, repetitions should be avoided once pointless (they are leg-work or, worse, busy-work that reduce interest in math and school, as I note from my own experiences), but to decide when an activity is pointless is tricky.
A better way can be to replace boring work on paper with some type of game, e.g. a computer game that requires correct answers (without “showing”!) to problems in order to score points, advance a character, or similar. Generally, this type of purpose-driven activity tends to be more entertaining and lead to a greater learning effect than “solve these near-identical problems because I said so”. There are limits to the benefits, of course, but any improvement can be welcome when it comes to (real or perceived) leg- or busy-work. (When I was a first grader, I tried to reduce the dreariness of repetition by re-imagining the numbers as armies and the arithmetic operations as battles. This moved me from extremely bored to merely very bored.)
Looking at arithmetic, e.g., someone who has learned how to solve problems like 4 + 5 = ? with ease might still be handicapped later, if he has not internalized the answers to very common cases. For instance, in a next step, calculating 4 * 5 = ? is much easier if someone simply “knows” that 4 + 4 = 8, 4 + 8 = 12, etc. Likewise, having internalized the multiplication tables makes later arithmetic much easier. (Which is not an endorsement of blind memorization of the multiplication tables, which schools often seem to push. The point is rather that using numbers to a higher degree, including more repetitions of multiplication problems can lead to such internalization without deliberate memorization. Here, the main topic of “showing” reappears: solving problems like 4 + 5 = ? and 4 * 5 = ? for the umpteenth time is much, much more palatable without the need to “show”—and the more so the closer someone is to internalization.)
It is true that this handicap likely remains mostly school-centric—for the great majority, it is a handicap in school, not in life. However, an effect of incomplete school-learning on life is quite possible and cannot always be predicted. For instance, when I learned basic German in school, I could solve exercises like finding the right form of the article for a preposition by reasoning and I stopped once I was reasonably confident that I could deduce the right form. (This was perfectly adequate for later tests and I did not, at the time, have an intention of moving to Germany—but I later did move.) However, because I stopped, I did not internalize the correct combinations, which proved an obstacle when I moved to Germany and had to arrive at the right combinations much faster than in a school exercise. In school, I could solve a problem like “gegen ? Wand” by first going through the right memorized list of prepositions (“durch, für, gegen, ohne, um” for accusative) and concluding that I needed the accusative, second strive to remember that “Wand” was feminine, third combine this with a memorized table of declensions to arrive at “gegen die Wand”. When talking to a German, however, this might have resulted in “gegen [looooooong pause], eh, ahem, die? Wand”. (This while the German had long internalized such a basic combination. Here, we also see that fluency is not just a matter of knowing, or being able to deduce, the right answer—but of doing so sufficiently fast. Here, very much unlike in school, an incorrect-but-rapid “gegen der Wand” might well have been the lesser evil.)
In 2025, I found an old exchange between me and a professor/TA/whatnot from Handelshögskolan tucked away in an equally old textbook. The exchange related to my disagreement with exam grading. A sticking point was that considerable points had been deducted because I had not used units in various calculations when showing.
In parallel with my main university studies, in engineering physics at KTH, I passed approximately two semesters worth of courses at Handelshögskolan, usually considered the premier business school in Sweden.
I left it at two semesters, even leaving a few started courses hanging, in part through my great dissatisfaction with the quality. (In part, for external reasons, notably, my exchange studies in Germany.) Some of that dissatisfaction shines through very clearly in the exchange, including that I explicitly complain to my counterpart about how both the test and Handelshögskolan in general had a focus on (to some approximation) rote learning over understanding—while understanding over rote learning was a core to my main studies at KTH. This to the point that some portion of my complaint refers to “sentence-specific questions” (“meningsspecifiska frågor”), i.e. that some questions dealt with the contents of specific sentences in the text book. (The details of this are not clear from the exchange, unfortunately. Swedish readers should note that the interpretation of “mening” as “sentence”, not “meaning”, is clear from the surrounding context of my complaint.)
As to details such as “professor/TA/whatnot”, I have to pass, the events being close to thirty years back.
This is interesting both as a parallel to, or special case of, showing and as a philosophical question—and even at 50, with two master’s degrees and far more math behind me than my then counterpart had, I am very sceptical to such deductions. (Indeed, I cannot rule out that I had more math behind me even back then, but neither can I guarantee it.) Now, without doubt it can be viewed as a pragmatic “best practice” to include units in calculations, simply to reduce the risk of errors (e.g. in that a factor has been forgotten from or spuriously added to a calculation or that a unification of two different units of the same dimension has been forgotten, say, by multiplying miles and kilometers as if both were kilometers). However:
In terms of “showing”, units are something that the brighter students (all other factors equal) might be more likely to leave out. Firstly, units increase the amount of work with (often) little return—especially, when the student has calculated in his head (often, then, implicitly with units) and is now transcribing as an act of leg-work. Secondly, a less bright student is less likely to be able to keep the correct calculations, roles of various numbers (and/or number–unit combinations), whatnot in his head. Thirdly, and partially overlapping with the next paragraph, a more mathematically advanced student/course/book/whatnot might focus less on units and more on dimensions, e.g. in that a particular integral is performed with regard to the dimension of time or the dimension of length, but not in seconds resp. meters, except, maybe, as a last step of just plugging in values in an otherwise completed calculation. (This often overlapping with a move from algebraic manipulation, which is often the main task to solve, to trivial arithmetic.) Indeed, a bright student might very well have an easy problem involving units and simply see what he puts on paper as plugging in values, as with an area calculation that involves plugging in the numbers 4 and 5 to get 20. (While either keeping track of units implicitly, knowing that 4 and 5 are in meters and 20, therefore, in meters squared, or doing a “unit calculation” in parallel, so that 4 * 5 = 20 and m * m = m^2.)
In contrast, an answer of “20”, instead of “20 m^2”, might be a legitimate cause for complaint, because it is not clear whether the unitless 20, as answer, is m^2, km^2, kg, or something else entirely.
An interesting perspective, and one that can affect the issue of showing considerably, is what the purpose of the on-paper calculations is. Firstly, a duller student might use these calculations as the “actual” calculations (or a transcription thereof from another piece of paper), while a brighter student might have handled matters in his head (entirely or with some reservations for just plugging in values) and just use the on-paper calculations for showing. Secondly, an implicit purpose is to explain to a later reader, a someone else, what the calculations were. Here, the brighter students might be more likely to fail, because they underestimate the need to communicate clearly with a potentially less bright teacher/TA/whatnot and/or a ditto who is used to less bright students and who, then, demands more stringency in showing, because he does not trust the students’ ability. (The high-school years can be particularly tricky here, as the students are closing in on their adult intelligence levels and the brighter will often already surpass the teachers, while younger years can see them too far from that adult intelligence and the college years often see a great increase in teacher intelligence relative high school.)
Outside calculations:
When I write a text involving measurements of the same unit, e.g. in the context of sports, I often say something like “all measurements in [unit]” and then leave units out. While the work saved might seem a trifle, there is, at a minimum, a psychological effect—and more real effects can add up.
In particular, there is more to the matter than just writing or not writing an “m” or an “s” here and there, including the need to pay attention to consistent use, e.g. that I have not accidentally given one long jump as “8.95 m” and another as “8.90”, let alone had a complete brain-fart and given “9.90 s” or made a typo for “8.90 n”.
In another area, it saves me a decision when and whether to use spaces (and, if so, what type; cf. below), when an abbreviation and when the full name of the unit, when a full name does or does not take a plural, and possible some other, similar, cases.
(Spacing is tricky, because it is not just a matter of good style but also of what capabilities are available in what context. Not giving a space at all is frowned upon but convenient. Giving a regular space risks an unfortunate line-break, while non-breaking spaces might not be available or require additional effort. A “thin” space often gives an optically superior result, but is, again, not always available—and do I really have the energy to dig up how to create the arguable ideal, a non-breaking and thin space, in the setting at hand?)
In terms of “philosophy”, the use of units in calculations borders on the nonsensical. For instance, 3 and 5 are numbers and can be multiplied to yield 15, but 3 meters and 5 meters are not numbers and cannot be multiplied in the same manner. For instance, I have often seen algebraic structures dealing with numbers but have no recollection of seeing one that combines numbers and units (but, in all fairness, they could exist without my having seen one). For instance, calculators and other tools for arithmetic take numbers, not number–unit combinations. At a minimum, we might need to move from arithmetic to algebra to justify units, but even that seems too thin to me—and it is hardly something of which the average teacher would even be aware (in favor of a “calculations are calculations” attitude). Possibly, math involving units is best seen as (what is often referred to by mathematicians as) “formal” and/or “symbolic” manipulations, short of (what many consider) “proper” math.
Some special cases can be argued. Consider percent when viewed as a unit. However, should percent be viewed as a unit? (Some convenience exists in terms of understanding the general type of information, e.g. that “51 percent of the vote” might be easier to understand than “0.51 of the vote”. However, at the end of the day, the unit or “unit” of percent is just multiplication by 1/100, and of a very different character than, say, the unit meter.) Ditto e.g. the unit mole, which could be seen as multiplication.
There are plenty of algebraic structures (and other math) that do not deal with numbers. Sets (and their associated operators) are among the most prominent examples. However, in my experience, these have much more in common with numbers than with number–unit combinations. Consider something as simple as multiplication on a field (even should the implication of multiplication on the field at hand be very different from school arithmetic and/or involve non-numerical elements). Multiplication would be defined over the elements of the field—but what happens if we want to use such multiplication to handle units? We could, maybe, define elements like 0 m, 1 m, 3.14 m, etc., in lieu of (or in addition to) 0, 1, 3.14, etc.; and could, then, handle meters. But we now have problems like 1 m * 1 m = what? If we take the result to be 1 m, we violate real world expectations (and might well see consistency problems); if as 1 m^2, we are faced with questions like what the result of adding 1 m and 1 m^2 is (addition of arbitrary elements being mandatory on fields); if something else, with other problems likely to follow. And what if we want to include seconds, e.g. to calculate a speed as 5 m/s? Because seconds are not present among the existing elements, we need new elements. Add new units, e.g. for electric charge, and further elements yet are needed—and since real-world calculations must be open to whatever units present themselves, even a very extended (pseudo-)field with units might be frustratingly limiting. Better then to view addition and multiplication as operations on, e.g., unitless reals, with units considered separately.
The above also points to the importance of dimension over unit. If we had an algebraic structure (other than a field) that did deal with meters in a satisfactory manner, adding seconds might be problematic, but adding inches would usually be trivial—inch and meter have the same dimension and, just as with percent above, only a trivial scaling is needed to turn a (real) measure in the unit of meter to an equivalent measure in inches or vice versa. (Problems could ensue in a situation where the algebraic structure was limited to e.g. integer counts of meters, but the practical value of something so limited seems low to begin with.)
An interesting special case is use of variables and whatnots in computer programming. It certainly is possible (often advisable) to handle variables in a “with unit” manner (e.g. by using a suitable object or, outside object orientation, a data type which combines a numerical value with a unit by using two internal variables); however, most of the time, variables that refer to a value with a unit only do so implicitly, e.g. in that 5.23 m/s is stored as the number 5.23 in a variable documented as using the unit m/s—and, likely, with the unit being secondary to some higher level claim about the nature of the value, say, that it is “the current speed of [whatnot] in m/s”. (With similar remarks applying to e.g. functions.)
In a program with such variables, an expression like x * y * z would then indeed amount to (hypothetically) 4 * 3.55 * 2.333—not (again, hypothetically) 4 m * 3.55 kg * 2.333 s. For a manual evaluation, it would be an outright error to include the units during calculation. It is, of course, important to pay attention to units to the degree that a later use of the calculated value is correct, e.g. that it is assigned to a variable with the implicit unit of m * kg * s and not (without prior conversion) e.g. m * lb * s. This, however, is virtually exactly the situation seen in various real-life plug-in-the-values and/or multiply-x-y-and-z situations.
However, in fairness to Handelshögskolan, I did have a similar issue once at KTH: A professor in a math course had spoken about how the bar over identifiers of vectors was just a pointless typographical convention (or some such—another event from the mid-1990s) and claimed that we were better off leaving the bar out. Initially, I took his advice to heart, but, shortly thereafter, I had a written exam in mechanics (?) and saw one point per question deducted for ... failing to put bars over my vector identifiers. (Unlike with Handelshögskolan, this was the sum of all deductions and I still got an A, so I did not pursue the matter.) To some degree, I see the professor as in the right, as it truly is a typographical convention, and one, arguably, of less interest than the question of units or no units; to some degree, I would, today, fault him, because it is not only quite easy to lose track, but can be tough on the reader; to some degree, I might make the correct answer contingent on the context, in that he might have more of a point when dealing with, e.g., abstract algebra than with a practical engineering problem; to some degree, I might claim that he (or I, as an adapter) was correct in idea but pragmatically unwise. Interestingly, however, without this exposure to more abstract mathematical thinking, I would not have left the bars out—fresh out of high school, e.g., I would not have done so and I would not have had these point deductions. Ditto, presumably, had I been a duller student who had taken less advanced math courses, had not seen the professor’s point, or otherwise had not been moved to follow his advice.
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