Renaming Mathematical Diseases

University of Saskatchewan

Self-*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.

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) = *x*^{2} + 3*x.* 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

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!

Ask any upper-*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-*x* + *y*)^{2}. Instead of FOILing (another issue for another time), which would result in *x*^{2} + 2*xy* + *y*^{2}, students, apparently riddled with fear, instead apply the square to each of the terms in the brackets, resulting in *x*^{2} + *y*^{2}. 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 *x*^{3} + *y*^{3}, 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

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 (*x*^{10} + *y*^{10}), 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} = *x*^{2} + *y*^{2}. 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

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* 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, *a*^{4}*a*^{7} = *a*^{11}), naturally (to me), I extended the session by asking the student to simplify (*a*^{2})^{4}. And there it was, sumonia, in all its glory: (*a*^{2})^{4} = *a*^{(2+4)} = *a*^{6}. This time, however, in what many consider as the highly coveted one-

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 (*x*^{2})^{3} = *x*^{2}*x*^{2}*x*^{2} = *x*^{(2+2+2)}. They understood that *x*^{2}*x*^{2}*x*^{2} = *x*^{6} but they’d understood that from the beginning. The issue I was facing, then, was how to explain that (*x*^{2})^{3} = *x*^{2}*x*^{2}*x*^{2}. As is customary, I attempted to draw upon what they already knew. So, I asked them to explain what an expression such as *c*^{4} 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 *c*^{4} 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 *c*^{1}*c*^{1}*c*^{1}*c*^{1} = *c*^{4}. He replied, “Yeah, I get the whole imaginary 1 thing, but I just don’t see why *c*^{4} 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-*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-*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 (*a*^{4})^{2} as *a*^{6} 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.

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-*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 conditions*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 *functionitis*

Chernoff, E. J. (2020, Summer/*The Variable: An SMTS Periodical,* 5(2), 41–*SpecGram* CLXXXIX.1.)