Inter-Lingual Personal Script—Daniel Swanson SpecGram Vol CLXXXIX, No 2 Contents Scriptominoes I—Trey Jones & Keith Slater

Extremely Amateur Mathematics Education Morphology:
Renaming Mathematical Diseases

Egan J Chernoff
University of Saskatchewan

Self-isolation during the COVID-19 pandemic became a necessary way of life for months (at the time of this writing). And, while being stuck at home has undoubtedly tested the human spirit, it has also inspired some spectacularly silly pastimes (e.g., running full marathons on very small balconies, marble racing and many others). Looking to contribute to the coronavirus corpus of distractions myself, this article builds upon Chernoff (2020), where I renamed certain mathematical diseases associated with common mathematical mistakes. So, as we look forward to, eventually, saying goodbye to the global pandemic, and having just said goodbye to sinusitis and functionitis in my last article, it’s now time to say goodbye to squaranoia, sumonia, and, perhaps, even logarrhea, which is where we begin.

Logarrhea: log(a + b) = log(a) + log(b)

The word association for logarrhea, from every last one of those people that I’ve asked over the years, is, of course, diarrhea. Unfortunately, all of us are familiar with diarrhea in some way or another, so there’s no need to elaborate on that condition here. The word diarrhea, if you didn’t know, is the combination of the prefix dia- (meaning through, throughout, or completely) and the suffix -rrhea (meaning flow, discharge, or secretion). Stemming from diarrhea then, logarrhea is considered the flow, discharge, or secretion of logs (logarithms). I must admit, given the familiarity of the term diarrhea, I am a little hesitant about changing the name for mistakenly thinking log(a + b) = log(a) + log(b). Maybe, just maybe, there’s a better option.

The term logarrhea works (arguably) because, in the mistake in question, it appears that the logarithm on the left side of the equation, log(a + b), flows (or discharges, or secretes) throughout or completely through the brackets to each of the terms in the bracket, log(a) + log(b). However, although both notions found in the term diarrhea (‘throughout’ from dia- and ‘flow’ from -rrhea) are both used in logarrhea, only the suffix, -rrhea, makes its way into the term logarrhea. Any attempts at a quick fix, such as dialogarrhea or logadiarrhea or some other combination, simply don’t work from a phonaesthetics perspective.

Looking more closely at the mathematics of the mistake currently associated with the term logarrhea, one could argue that students are distributing the logarithm through the brackets, much in the same way that they were earlier taught to distribute both numbers and variables in front of a bracket, such as in 2(x + 2) = 2(x) + 2(2) or in x(x + 3) = x2 + 3x. If this is the case, then they are applying what they’ve already learned to a novel scenario. The only problem is that applying what they’ve learned, in this case, doesn’t work. In line with this reasoning, then, the root of the problem has less to do with logarithms and more to do with incorrect distribution or the distributive property.

Full disclosure: While isolating at home during COVID-19, I spent a lot of time attempting to rename logarrhea. I tried to, somehow, integrate the prefix sinistr/sinistro- (meaning left or left side) and dextr/dextro- (meaning right or on the right side), to no avail. I focus-grouped a bunch of terms only slightly different from logarrhea; however, logarrhage (utilizing the suffix -rrhage, meaning to burst forth) and logarrhexis (using the suffix -rrhexis, meaning rupture) remained ripe for the same critique I have just presented for the original term. And then, one day, on a long walk trying to kill time, it hit me. As Crystal Pepsi, New Coke, and other infamous rebrands have taught us, don’t mess with a classic. And so I realized, after much deliberation, that logarrhea must remain logarrhea.

As I continued walking, I also realized that my earlier critique regarding logarrhea might not hold as much water as I initially thought. Yes, logarrhea, which stems from log- (denoting logarithms) and -rrhea (denoting flow), does not integrate dia- (meaning through, throughout, or completely). However, the accepted term is logarrhea, and not logrrhea or logorrhea. In other words, and remembering that it’s extremely amateur math ed morphology taking place here, it could be argued that the -a- between log- for logarithms and -rrhea for flow is actually from the dia- prefix. A piece of a prefix, if you will, which I’m sure has its own term. And there you have it, the word logarrhea stays, and there’s good enough reason (in my mind) for it staying. Logarrhea, then, becomes the first of the named infamous math mistakes that stays. Classic!

Squaranoia: (x + y)2 = x2 + y2

Ask any upper-years math teacher: Squaranoia is a thing. Initially, they might not know what you’re talking about, because squaranoia is also known as brackephobia, or brackaphobia (portmanteaus of bracket and phobia). Both terms work, and both have a rather nice ring to them. However, as we dig a little deeper, we’ll see that perhaps it’s time for a change.

There are several issues with the term squaranoia. To the best of my knowledge, squaranoia is rooted in the term paranoia. In a general sense, the notion of paranoia (i.e., anxiety- and fear-riddled thoughts that can lead to irrational decisions) is referring to what should happen versus what does happen when students are confronted with an expression such as (x + y)2. Instead of FOILing (another issue for another time), which would result in x2 + 2xy + y2, students, apparently riddled with fear, instead apply the square to each of the terms in the brackets, resulting in x2 + y2. In other words, squaranoia implies a fear of properly squaring. Phonaesthetically speaking, squaranoia is pretty good, but from an extremely amateur math ed morphology perspective, perhaps things could be even better.

Paranoia, to the best of my knowledge, is derived from the ancient Greek words para (irregular) and nous (thought, mind). As for squaranoia, the previously undocumented prefix of squara- would have to mean something along the lines of ‘of or related to squares or squaring.’ However, there’s the issue of increasing powers. For example, when a student rewrites (x + y)3 as x3 + y3, is the student suffering from cubanoia?! Similarly, then, students might also suffer from quatranoia, quintanoia or pentanoia, and so on. However, as seen in each of the examples, no matter the exponent outside of the bracket, the error in question is essentially the same. In other words, squaranoia, cubanoia, quatranoia, and pentanoia, are really one and the same issue, which is why brackaphobia or brackephobia is arguably a more accurate term (and a great term, phonaesthetically speaking). However, that doesn’t mean we need to stop at brackephobia. Let’s now look at things more closely from a morphological angle.

The term phobia, as you are undoubtedly aware, refers to an intense fear or aversion to something. Akin to how arachnophobia describes an intense fear of spiders (or arachnids), brackephobia would, similarly, indicate an intense fear of brackets. Now, phobia concurrently exists as a standalone word and as (what I’ll call) a loose suffix, and because of this, many people take artistic license. For example, I have a phobia of being buried alive, and so I might say, to my friends, that I have being-buried-aliva-phobia. In fact, the proper term to describe this fear is taphophobia. To give another example, someone who is afraid of golf might think they suffer from golfophobia (or golfaphobia or golfephobia), when in actuality, they suffer from golfphobia. Based on the former and latter then, brackephobia or brackaphobia (or even bracketphobia) would work better as a general mistake descriptor than, say, squaranoia or cubanoia.

By changing terms, there is no longer a need to be concerned with the particular exponent outside the brackets, as the issue now lies with the brackets and not the exponent. Recognizing what is supposed to happen in these instances, higher powers should result in more brackets: For example, (x + y)5 is equivalent to (x + y)(x + y)(x + y)(x + y)(x + y), which is akin to encountering more and more spiders, which should result in a greater degree of fear. At the same time, though, the mathematical errors resulting from unawareness of necessary brackets, leading to, for example, (x + y)10 being simplified to (x10 + y10), would only freak someone out if they realized how many brackets are actually involved. While there may be fear, even for me, associated with correctly simplifying an expression such as (x + y)(x + y)(x + y)(x + y)(x + y)(x + y)(x + y)(x + y)(x + y)(x + y), it’s time to dig a little deeper into the almost accepted term brackephobia to see if there’s an alternative to address issues raised.

Without getting into the larger question of whether you can be afraid of something you don’t know exists, I am recommending brackephobia over squaranoia. (I should point out that, yes, you could fear that a monster exists under your bed, but perhaps the notion of a monster under your bed had to be planted in your head before you could start worrying about Gary. And, yes, I guess that we do often fear the unknown.) My point is that the notions of paranoia and phobia aren’t the best descriptors of what is going on when a student makes the mistake that (x + y)2 = x2 + y2. After all, many who make the error don’t even know that (x + y)2 is equivalent to (x + y)(x + y).

What remains, then, is a way to describe not a fear or a phobia, but rather what’s happening when expanding an expression such as (x + y)2 (and, if possible, to use a medical term to keep with the theme of diseases). I spent a considerable amount of time searching medical terms to describe the fear of not being aware of, or not understanding the unknown (e.g., panphobia), to no avail. Almost on the brink of having to accept brackephobia, while scanning one last time through a list of suffixes and prefixes, I found a few items that I had previously dismissed too quickly. With many terms available to build upon (for example, brackets, parentheses, powers, and exponents), with suffixes such as -staxis (dripping or trickling) and -ptosis (falling, downward placement), and with prefixes such as cata- (down) and acr/acro- (extremity or topmost), I found myself with renewed confidence in my ability to rename squaranoia and brackephobia.

Without further ado, I contend that expoptosis (rhymes with halitosis; the second p is silent), defined as the falling or downward placement of exponents, should replace squaranoia and brackephobia on the list of mathematical diseases. Two key elements are captured in the word expoptosis. First, for the reasons detailed above, the notions of phobia or paranoia are removed from the term. Second, the term describes what is actually taking place during the mistake. In other words, the exponent “drips” or “trickles down” to each of the terms inside the bracket. As we’ll now see, a focus on the exponents will also be at play when I attempt to rename sumonia.

Sumonia: (ax)y = a(x+y)

Sumonia was a prevalent problem in the math classes that I used to teach, but I was better able to get a peek behind the curtain of what some students might be thinking in the sumonia scenario thanks to a tutoring session I had on the same topic. Having just ripped through the topic in class at school one day, I thought I would draw on the same material I had used earlier that day to “inform” my tutee that same evening. Not so fast, I would learn.

After a session about simplifying powers, I didn’t think twice about extending my lesson to what was presented in that particular textbook as the “power of a power rule.” Having just finished several examples involving adding and subtracting exponents (for example, a4a7 = a11), naturally (to me), I extended the session by asking the student to simplify (a2)4. And there it was, sumonia, in all its glory: (a2)4 = a(2+4) = a6. This time, however, in what many consider as the highly coveted one-on-one setting, I was sure I’d be able to help my tutee deal with their mathematical disease. Again, not so fast.

My tutee had some pretty solid arguments for why they did what they did. “Well,” they began, “when you have the same base, you add the exponents.” To this, I had no immediate retort. “There’s only one base in this question, which means that you add the exponents,” they continued. In response, I fumbled around with different bases to make a point. I flailed using different letters, different numbers, symbols and even pens from the pencil case on the table for some reason. Nothing landed. No matter what I did, my tutee could not see that (x2)3 = x2x2x2 = x(2+2+2). They understood that x2x2x2 = x6 but they’d understood that from the beginning. The issue I was facing, then, was how to explain that (x2)3 = x2x2x2. As is customary, I attempted to draw upon what they already knew. So, I asked them to explain what an expression such as c4 meant. They just sat there. With the tutee’s parents within earshot, uncomfortable with the silence that was filling the room, I said, “Well, I know that you know that c4 is the same as c⋅c⋅c⋅c.” My tutee replied, “Yeah, I never really got that.” After another awkward pause on my part, I wrote exponents of 1 above the c’s to show that c1c1c1c1 = c4. He replied, “Yeah, I get the whole imaginary 1 thing, but I just don’t see why c4 is c⋅c⋅c⋅c.” In my head, I was scrambling for how to explain something that, to me, was so obvious. After the session was over and I skulked out of my tutee’s house (cash in hand), I began to question the term sumonia.

I’m confident, and stand to be corrected, that the mathematical disease called sumonia is derived from the term pneumonia. Pneumonia, as we’ve become more acquainted with during the COVID-19 pandemic, is an infection of one (or worse, both) of the lungs caused by a bacterium, virus, or fungus. In the corresponding mathematical mistake, one adds, that is, sums, the exponents, as opposed to multiplying them. Summing and pneumonia, then, becomes sumonia. Keeping with other names thus far, let’s examine the quasi-medical morphology of sumonia. Speaking again as a morphology amateur, pneumonia is derived from pneumon- (lungs) and -ia. However, the term sumonia suggests that pneumonia is actually a combination of pneu-, which is not necessarily a prefix, and -monia. It should be pointed out that, yes, there are many words that end with -onia, but this is not, from what I’ve gathered, a medical suffix. As such, I contend that the term sumonia is ripe for replacement.

When looking to replace sumonia, it is important to call attention to the fact that the mistake is occurring at, for lack of a better descriptor, the level of exponents. As a result, I propose tacking on a medical prefix or a suffix to the word exponentiation (loosely defined as the mathematical operation involving raising a base to an exponent, which is really what should be going on when simplifying) to replace sumonia. When going through possibilities, I considered some adequate prefixes, such as iso- (same), ite- (resembling), and peri- (surrounding or around another), but two fixes immediately came to the forefront as leading contenders.

Before declaring sumonia’s potential replacement, I had to make a difficult decision between using the prefix pseudo- (false or fake) and the suffix -oid (resemblance to). Pseudo­exponentiation, that is, fake exponentiation, works rather well because pseudo- is such a well-known prefix and, with respect to the mistake, it could be argued that fake exponentiation is what is taking place. The other possible alternative exponentiatoid, the alternative that I have chosen to replace sumonia, focuses not on the process but on the product that results from making the mistake. For example, should an individual mistakenly simplify (a4)2 as a6 then that person’s pseudo­exponentiation has resulted in an exponentiatoid. A little hard to pronounce at first, expo-nen-she-a-shoid rolls off the tongue, eventually.

Changing Terminology

To borrow a tired but true phrase: Change is the only constant. During the time of COVID-19, for example, the term social distancing was replaced with physical distancing. The latter term, in theory, was adopted to help us better keep our distance to reduce spread of the coronavirus. Along a similar vein, also during the time of COVID-19, in this article and in Chernoff (2020), I have proposed a rather radical renaming of mathematical diseases, mathematical diseases infamous enough for their own monikers. Gone, but not forgotten, are the terms: sinusitis, functionitis, squaranoia and sumonia; in their place, respectively, I propose we adopt the more quasi-medically and quasi-mathematically accurate terms of lateral­parenthetic­sinu­centesis, endo­parenthetic­function­ostomy, expoptosis and exponentiatoid. Of course, as we all know, change begets change, which raises the notion of mathematical diseases.

As I also recently noted, the disease found on the list of infamous mathematical diseases should, rather, be called mathematical conditions. Yes, there is a certain sense of shock value the first (few) times you hear of “mathematical diseases,” but mathematical conditions, from a variety of angles, is more appropriate. First, “we live in rather sensitive times now, the days of declaring that students are riddled with various mathematical diseases are probably over” (Chernoff, 2020, p. 43). Second, parsing diseases and conditions allows for the further incorporation of symptoms and syndromes (i.e., a set of signs and symptoms, correlated with each other, and often associated with a particular disease or disorder).

Long, long ago, when the poster of mathematical diseases hung prominently on the front wall of my math classroom, I would not hesitate to let a student know that they were riddled with a mathematical disease such as squaranoia, logarrhea, sinusitis, functionitis, cancellitis, sumonia, rootobia, negativitis, or moveitis. Something about the disease angle landed with students and they often never made the mistake again. Some would laugh, but not all, which brings me to present day. I would not recommend singling a student out in your class and telling them they are riddled with a particular disease. Things are different now. Alternatively, by embracing new terms, as mathematical conditions and not diseases, you could let a student know that lateral­parenthetic­sinu­centesis and endo­parenthetic­function­ostomy are mathematical conditionsabnormal states of mathematical health that interfere with usual mathematical activities like simplifying. How far you wish to get into parsing mathematical diseases, symptoms, conditions, disorders and syndromesfor example, the parecentesis syndrome (i.e., the puncturing and draining of brackets) can be found at the root of many mathematical conditions including (but not restricted to) logarrhea, sinusitis and functionitisis a matter of how far one wishes to take this COVID-19-inspired contribution into a math class.


Chernoff, E. J. (2020, Summer/Fall). Reducing Inflammation. The Variable: An SMTS Periodical, 5(2), 41–44. (Republished as “Renaming Mathematical Diseases: Reducing Inflammation,” in SpecGram CLXXXIX.1.)

Inter-Lingual Personal ScriptDaniel Swanson
Scriptominoes ITrey Jones & Keith Slater
SpecGram Vol CLXXXIX, No 2 Contents