Mathematical Jargon

Mathematics is a subject in which the jargon is required to be especially rigorous and unified. Like other bloggers this month, I will discuss how jargon, terminology, and other wordly matters play out in mathematics.

A group
of kids [cc]
A group in
mathematics [pd]

One of the first important distinctions in mathematics is that certain words are used as in normal English, while others are used with specific technical meaning. For example, the term group is used in common parlance to denote some specific set of objects, people, etc. that have some common characteristic or purpose. In mathematics, a group is a particular type of mathematical structure. Mathematicians use different words (like set, family, or collection) to refer to groups (in the common English sense) of things that are not endowed with this structure. It is even important to split hairs when using such words. The term set is defined in a specific way in mathematics, but it means precisely what it might mean in English. However, there are restrictions on what a set could be. Russell’s Paradox is one example of how something that could be called a set in English is not a set in mathematics. And so a new word is created to describe this kind of thing – it is not a set, it is a class. Fellow GSNB blogger Michael notes here the confusion that may arise if common English terms collide with technical terms.

Mathematics often borrows or redefines terms from common parlance, from other fields of study, or even from different areas of mathematics. The important consideration is whether a reader would understand – generally erring on the side of caution, defining words and phrases as needed. Mathematical exposition is peppered with definitions in a way others may find only in dictionaries. But if an astronomer were to read about a syzygy in a physics paper that requires some mathematics of that sort, he or she would be confused without having a mathematical definition handy to differentiate this from the astronomical definition of syzygy. It’s hard to fight the urge not to define terms throughout this blog post!

An important quality of mathematical exposition is not just mathematical fluency, but clarity. Terminology should be used judiciously. This is important because the terminology does not just describe or expound the content, it is itself content. Carolyn, another GSNB blogger, discusses here the construction of meaningless strings that may sound very impressive if not read very carefully. This is true in mathematics too, where a randomly-generated paper was recently accepted for publication, albeit to a for-profit journal that seems to lack appropriate peer-review. And, importantly, anyone with mathematical training could have spotted this paper as fraudulent without any particular specialty or knowledge. It’s virtually impossible to “bluff” in this way in mathematics.

This graph has many potatoes[c]

Abbreviations, wordplay, and figurative terminology are all useful in mathematics, but provide further barriers to understanding. The term “subadditive” describes a function where f(x+y)≤f(x)+f(y). The “sub” gives you ≤ and the “additive” gives you the + sign. It’s a definition one can parse from the word itself, given experience with mathematical terminology. But if this were abbreviated, it would lose that meaning. For example, a directed acyclic graph becomes a DAG; a partially ordered set, a poset; a universal Turing machine, a UTM. A separable completely metrizable topological space is called a Polish space (because they were first studied by Poles). Similarly, a ring without an identity element is called a rng (the i is removed, since there is no identity). And if acronyms and wordplay were not enough, terms like “potatoes” and “squiggles” can be used to describe something with a more precise meaning in an informal context. Mathematicians describe structures as “well behaved” or “badly behaved,” and even the term “almost always” has a precise, technical meaning. There is no snake involved in the snake lemma.

In the end, mathematical terminology is important because these terms, ideas, structures, etc. make up the building blocks of the theory. But they should be explained clearly, concisely, and precisely, and so with the audience in mind. Technical language should be an accessory to normal language, allowing us to make more refined and meaningful statements. Mathematical jargon allows us to use technical (and even non-technical) language to make progress, not to obfuscate it.

Images used in this entry are used under fair-use and/or under licensing guidelines set forth by the copyright holder that allow use in this blog, as presented for educational or critical commentary. Images are copyright their respective holders and credit or source is indicated in each caption or in the text of this entry, as applicable.

Teaching Non-Majors

One important aspect of being a teaching assistant is learning to teach non-majors, since in many cases, these students don’t come to class with a strong interest in the subject or with particular or special motivation for the course (it is, after all, not in their major subject). In my experience in mathematics, I have seen that the plurality or majority of teaching resources seems to be spent teaching students outside their respective department (at least by some measures, e.g. number of courses offered). This is probably true of many other departments. Teaching majors being a serious and core priority, teaching non-majors should nonetheless be a different, but still important, sort of priority.

An important factor in teaching non-majors is identifying the goals of the course. Generally, saturating students with content is how most syllabi and curricula seem to look on paper, but when I teach a calculus course, I know that our major goals are to build mathematical and quantitative literacy, develop the skills involved in calculus, and give students the required background for their majors and for their careers. This is universal, independent of the intended audience (biological sciences, social sciences, engineers) or the level (we have 4+ semesters of mathematics for non-majors, depending on their curriculum). Quantitative literacy is an important goal of mathematics education, and is a reason mathematics is a component of many majors (and of other general requirements). As Michael, another fellow blogger, mentions in his recent post, scientific literacy (and I would say quantitative literacy, statistical literacy, and other such matters) are important for our civil discourse and our society in general.

It is important for non-majors to understand expectations, especially expectations surrounding assessment. Alexandra mentions this in her post this month. Student work should be legible and comprehensible – this is very important in mathematics I can say from experience. Establishing the expectations and assessing students fairly, but firmly, makes an assessment tool more effective (and easier to grade not just in itself, but by soliciting good responses from students). Remember that this is not a non-major’s “native language,” so to speak.

Brian mentions in his latest post that sometimes students are hopelessly out-of-touch. That is certainly the case, but when teaching non-majors (or introductory classes, or interdisciplinary classes) it is especially important to adapt to students’ interests and abilities – otherwise, they are indeed pushed more and more out-of-touch. There is usually a reason students are required to take a course, but they don’t necessarily see it that way. Many students (freely!) confess that courses are often things to “get out of the way” – if a lecture, quiz, homework set, or discussion can develop their interest and give them some hands-on time with the course material, it may spark interest and make the course meaningful and connect them better to goals like quantitative literacy (or a respective equivalent).

Fellow-blogger Jennifer speaks about the enthusiasm of TAs in her most recent post, and to tie that discussion into this post, I would assert that non-majors do not usually share that enthusiasm. It is important to identify the level of interest students have, and if there are enthusiastic students, give them opportunities to extend and enrich the course. But if, as is likely, the majority are not especially interested, it would be a mistake to disconnect from students by expecting them to connect with that level of enthusiasm. Not that enthusiasm is bad (it’s great!), but it’s important to meet them at their level – and also to meet them at the interface of the course and the topics about which these students are enthusiastic.

Research in Mathematics

Working in mathematics, I’ve found myself often asked the question “What do you do?” Sometimes the expected response is my “elevator pitch” (the short blurb about my area of expertise). But sometimes the question is more basic: “What is it you do, though? Do you just sit all day and think?”

Thinking [cc]

Now, to a large extent, many people in research spend all day thinking. However, mathematics is not simply the art of staring at a problem until the solution materializes in one’s head. (It’s worth a try, but often solutions do not come from epiphany alone.) I would like to discuss a few of the ways in which research is conducted in mathematics, with emphasis on the parallels and similarities that may exist between mathematics and other fields, perhaps to somewhat debunk that notion there may not be any such similarities.

Nature [cc]

Mathematics research revolves around proving new theorems — mathematical statements that can be deduced from the fundamental axioms of mathematics and from preexisting theorems. Generally, though, the procedure is not to make a big pile of the existing statements and to try to string them together randomly until one forms a coherent deduction that results in something meaningful. That would be pretty rough sailing! Mathematics relies on conjectures, put forth as believed to be true and hopefully proven by someone at some later time. While there are conjectures (e.g. Goldbach’s conjecture) which remain unsolved for long periods of time (sometimes resulting in notoriety), most theorems start out as rough ideas or propositions that are developed with increasing structure and refinement until they are proven. In addition to proving new theorems, other steps forward in research include constructing examples of mathematical structures and verifying theorems by re-proving them in new ways. Computational work is also done to improve theorems in the case that a theorem is quantitative (or sometimes, to prove that a quantitative result is best-possible).

El. J. Comb. logo

Know the literature: As in most fields, the mathematical literature is vast, and perhaps especially in mathematics, it is easily accessible. Increasingly, mathematics journals are available online — not just through the library system, but free for instant download on the Web. Having less concern for the preservation of intellectual property, many editorial boards have shifted to such open/free publication. (Indeed, I myself have a publication in The Electronic Journal of Combinatorics, which is precisely such a journal.) There is also the arXiv (the X is pronounced sort of like χ, the Greek chi), which hosts preprints of papers and other works of mathematics (and many other fields).

Being familiar with the body of literature, both seminal papers and other older works as well as the current cutting-edge work (as it appears first, usually on the arXiv), is an important part of conducting research in mathematics. Jacob Fox came to Rutgers in 2009, when he was at Princeton, to speak at a seminar. He noted during his talk the importance of being familiar with the literature, mentioning in particular how his knowledge of a certain publication helped him and his coauthors solve a problem.

Mathematics [cc]

Crafting and Proving Good Conjectures: One of the more important questions is where to start — if we’re going to prove a statement is true, what is that statement? Generating good conjectures is not a matter of guesswork or divine inspiration, at least not entirely (although the former may have helped from time to time, and the latter is open to some debate at least). Increasingly, experimentation is a common way to generate conjectures. It is also often useful to test conjectures in small, typical, or special cases (where “small,” “typical,” and “special” depend on the problem at hand). Usually a conjecture applies to too many cases to test them all (sometimes, infinitely many cases), so this methodology is often used to verify that the conjecture is sometimes true, but not to verify the conjecture exhaustively. (Conversely, experimentation may lead to a disproof of a conjecture by identifying, constructing, or otherwise elucidating a counterexample.) Experimentation may also help unearth components of the proof of the conjecture at hand.

It is also crucial to have a firm understanding of the big picture in the field where these questions are being asked. There is a substantial amount of context and content that guides someone to the right kinds of conjectures and the proofs of those conjectures. Mathematics is a field in which the objects of study are highly structured, and knowing these structures helps eliminate some of the technical clutter that can obfuscate the underlying truths that one wishes to prove and the bits and pieces that go into proving them. Many proof techniques can be adapted to different situations, so in some sense theorems may be proved by matching a generalized proof to a statement you would like to prove specifically.

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Building Theories and Solving Problems: Tim Gowers is famously credited for roughly dividing mathematicians into the two categories problem-solvers and theory-builders (or rather, he is credited for noting this division in his oft-quoted The Two Cultures of Mathematics). I won’t discuss this dichotomy, but these two activities characterize much of the research done in mathematics. Proving single, unrelated theorems one-by-one is not usually how research goes. Rather, the enterprise involves longer strands of investigation — a dozen theorems sometimes collapses into a single stronger and better statement after enough exploration and refinement. Meanwhile, single ideas branch into many avenues of investigation. But generally, the aim is not to knock down one theorem, then turn around π radians and start over, but to work on larger-scale investigations. I could make a metaphor about bowling pins or dominoes, but I think the idea is clear. An important aspect here is also collaboration, which is a major element of research for many mathematicians. Working on papers is one part of collaboration, but other important activities include seminars & conferences (as participants and as organizers), expository writing, editorial work, and many other collaborative activities.

Structure [cc]

So the venture is to find good lines of research and establish some clear, path along that line. There are the two approaches. The first is to identify important problems and build up theory to solve them. One famous example is Fermat’s last theorem, which conjectured centuries ago and recently proved by Andrew Wiles. During those centuries, large swaths of mathematics were developed in large part as attempts to prove this conjecture (including Schur’s theorem, one of my personal favorites). This “problem solver” work weaves what might be the leading strands of the theory, loose and rough but pushing outwards farther than neighboring strands. Such work often moves mathematics in innovative or interdisciplinary directions, building bridges between fields of mathematics, and may also connect with work in applied mathematics. The “theory builders” weave strength and cohesion into the fabric (to extend the metaphor). To this end, they focus their research on developing and enriching the theory. They may work to classify all types of a particular structure, for example. Such work includes that of several Rutgers faculty in classifying the finite simple groups. This theory-building reinforces others’ work as they develop and solve conjectures, as it makes the underlying theory more robust.

Images used in this entry are used under fair-use and/or under licensing guidelines set forth by the copyright holder that allow use in this blog, as presented for educational or critical commentary. Images are copyright their respective holders and credit or source is indicated in each caption or in the text of this entry, as applicable. Thanks to Yusra Naqvi for her helpful comments and suggestions.

I can just Google it.

Despite having several thoughtful blog entries “in the works,” I thought I’d make my first post about something perhaps slightly amusing and somewhat observational. We live in an age well past the dawn of the internet. Indeed, I would not call it the age of information, but rather, the age of data. Social media, bulk email, Youtube, cell phones, smart phones, Twitter, the blog-o-sphere, and everything else — we are highly connected to media. I was struck recently when a group of undergraduates was somewhat shocked that I knew some basic theorems of mathematics off the top of my head. And while that was surprising to them, they were thoroughly confounded when I identified an arachnid as something other than a spider  — not only that it was possible to identify such things by their physiology, but that one could do so without aids or notes (some were also unaware that such creatures exist at all). Indeed, my observations and conclusions were checked on Wikipedia as soon as they got to a computer.

I don’t know if there are technical definitions of the terms data and information (and whether those definitions vary in whatever fields they find use), but to me, they have sharply different connotations. I believe the superhighway of the Internet, and all of its major repositories of (mostly) text-based media, are not conductors of information, but rather, of data – data of varying types, formats, detail, and reliability. And for that reason, significant research is being done to distill and interpret large sets of data, in myriad formats and structures and scenarios (which is not the topic of this blog post).

What I wanted to discuss is how “looking it up” has become such a pervasive technique for the acquisition of information, and why — with so much data around — it is important to know precisely what this process really means. In the end, I think there is an important distinction between looking up information and hearing it from an authority (in a lecture, discussion, conversation, correspondence, or however else). In person or by some personal medium of communication, knowledge and insight can be expressed and even transferred. The ideas are filtered and interpreted carefully, especially in a dialogue or discussion, and the information is contextualized and is explained with greater depth and breadth than a Google search or a Wikipedia article might provide.

Indeed, I believe various tools like Google, Wikipedia, or Wolfram Alpha (for those of us who are mathematically inclined) have all changed the nature of our interactions (be we students, teachers, or those outside of the university setting) with information and data. Painful anecdotes circulate about students who complain that no sources exist for their term paper because Google can’t find any, or who complain that a math problem cannot be solved because Wolfram Alpha can’t solve it. If only research were so easy!

That misunderstanding, which may be a more subconscious sort of convolution of bad habits and lack of information about better practices, really limits students. And these same misconceptions bleed over into the younger generation in academics and the workforce — and even into older generations as well. Bad habits are hard to break, but surely, they are somewhat easy to adopt. What used to be somewhat novel has become the go-to method for trying to find information, but is it the best first-line for that process? If not, when are alternatives more appropriate?

The reason it is so easy to adopt this model — that all information and knowledge can be obtained by reasonable computer search, and thus does not need to be known or understood beforehand — is that for trivial or logistical information, it has become increasingly valid. Indeed, I am blessed with a relatively uncommon name, and the number one Google search terms going to my website are things like “Kellen Myers Rutgers office hours” or “Kellen TA Math” or the like. Students who need logistical information about my office hours, course policies, etc. can find my website and find all that information there. But if they need help with the course, they should not Google “Kellen Myers calculus homework answers.” I doubt this would be useful, and at the very least, I don’t recommend looking up homework answers online to any students.

Once, in particular, I was sorely disappointed to find students asking (many of them repeatedly) when office hours were. Finally, when one student asked by email for the second time (having forgotten my previous response?), I responded that this information can be found by a Google search or by visiting my website — the student was very displeased, and even accused me of disrespect and dereliction of my duty as a TA for not answering the question directly. This stance, in addition to being somewhat hyperbolic, is an unfortunate passing over of the resources and information at hand. Students can often find a wealth of information about their courses’ logistical information, about their instructors’ availability, library hours, school policies, etc. etc. There is a huge amount of information out there! Here, by the way, is an important note for those who provide this information — doing it correctly, effectively, and clearly is an important part of the administrative side of instruction. Five minutes putting together a clear, concise webpage for a course may save hours of emails, confusion, etc.

But information like times, dates, locations, birth-dates, and so forth, seem to be easily accessible and, if the context is understood, a precise online search would yield this sort of information easily. There is no problem discerning from a search what data are valid and which can be trusted to give the correct, valid, desired piece of information. For example, if I needed to know offhand what year the Magna Carta was issued, without the Internet I could (very cautiously) ballpark it as 1100-1400, but a Google search brings it up immediately (the first hit being Wikipedia, which has the information right there in the first paragraph). But knowing the context helps, as a similar search attempting to find the year Marie Antoinette gave her famous “Let them eat cake.” speech brings up several news stories about Mitt Romney, various complicated historical accounts of how she never actually said such a thing, and much more data (related and unrelated) than I would have liked. More knowledge and context might help me sift through that information, but here Google does seem to fail to deliver precisely the datum I was expecting to find.

In the long run, this issue has an impact on how we teach students to find information, be it informally (that is, day-to-day stuff) or in some formal setting (e.g. term paper). This generation of undergraduates has, after all, never used an actual card catalog. Everything is an electronic search, but knowing how to search effectively and what to expect from various search tools is important, and this might be something students (and scholars, and others) lack. We may not have knowledge of the tools at hand, nor of the results one can obtain when using such tools (or how to use such results responsibly).

Indeed, upper level math courses in particular become a bit tough when planning homework. Is this problem solved online somewhere? Will my students find it through Google? It’s a pretty good argument against posting solutions when often, standard or important exercises would be rendered ineffective by having solutions available prematurely. Perhaps this is another piece of good practice for instructors, in both keeping solution sets off the internet and learning to adapt when such information becomes ubiquitous. It may be challenging, finding ways of still giving effective problem sets without running afoul of these online solutions, but I  would say it is usually possible.

The question becomes more complex when computers can solve problems too. In algebra and calculus, students can make use of Wolfram Alpha to solve problems (now with steps provided explicitly, which makes cheating-detection quite difficult). And this isn’t confined to homework or take-home assignments. Indeed, I have heard of students whose phones have been confiscated during exams, with the Wolfram Alpha App wide open with a solution to an exam question on the display. But is Wolfram Alpha the enemy? Let’s hope not — it’s an enemy we can’t fight! It won’t go away, and surely no one can believe such a service could be blocked, censored, or limited in some way.

Like Google or Wikipedia, Wolfram Alpha and any other such site will be there to provide students with access to various data, and how students use that data is something to which we must respond well, but also for which we can prepare. (And here, perhaps, I disagree with Wolfram’s description of its product as a “knowledge engine. I would consider it a data engine, and that usually it could be considered reliable enough to provide information, but not knowledge. To me, knowledge of a calculus problem is the ability to understand the methodology of the solution and solve it without an outside aid of that sort. I realize all three of these terms I have used without definition, and I am not brave enough to venture some postulated set of definitions for the terms despite using them freely.)

But, if students are taught how to effectively utilize searching resources, including things like library catalogs and journal resources, they will have access to a better base of data. If we prepare them to filter and interpret that data, we can mediate the problems created by the influx of data that might overwhelm someone searching on the Web. And if we prepare resources (mainly, websites) that provide important and essential information through search engines effectively, students will find the right information right away, using the resources that they have come to primarily rely on for acquiring data. And for times when this is not the right way to find data, we can help students learn to use other resources — which may, in this generation, be new to them (up until college, Google and Wikipedia may have sufficed entirely). Eventually, we can hope they will not be reliant on these resources for all data, as research, writing, learning, and other experiences should impart knowledge and information. And we ourselves, as faculty, graduate students, undergraduate students, or anyone else, can learn to better use these resources. Search engines and other online data/information resources can supplement instruction and research, and are incredible tools for data acquisition, but knowing when and how to use them is crucial — not only to prevent misuse or over-reliance on these resources, but to also make use of them as important and increasingly abundant tools for gathering and refining information.