Job Search Advice

Recently, I attended a series of workshops sponsored by the Graduate School and Project AGER, in which Dean Barbara Bender provided useful, concrete advice to students preparing for an academic job search. An audio transcript of one of these workshops can be found here. I was hoping to reflect on a few of the more interesting and/or important points from this workshop.

The series of two talks was hours long and filled with useful information, so I will highlight a few points, hoping to give some advice that is the most important and some that may be less common (since job-searching students are often inundated with redundant advice).

First, check the accreditation of a school to which you might apply. Make sure that the school is accredited by one of the regional accreditation boards. Some fields of study also require special accreditation — if this is true in your field, check for that too. Likewise, check whether the institution operates as a non-profit. While there are plenty of job opportunities at institutions that are not accredited and/or for-profit, it’s important to understand that aspect of the institution to which you might apply for a job.

It may also be important to consider the type of school (research-oriented, liberal arts, technical, big, small, rural, urban, religious, etc.), and to consider the school’s mission. Most schools have a mission statement (or something equivalent) that you can read. It might give you some idea about the guiding principles of the school’s educational and scholarly work and help you consider how well the school fits your career plans. The history of the institution may also provide some insight here, as might the status of faculty collective bargaining, benefits, and so forth.

You should also consider positions in schools besides the usual four-year institutions. Community colleges and other such institutions may often be overlooked in a job search, but some community colleges, 2-year institutions, etc., can offer competitive salaries — and often with tenure lines and a strongly unionized faculty. There are also jobs outside the tenure track, like contract teaching, online & continuing education, institutional research, and administrative work.

If your career goals include working in academia, but you leap into industry, it is possible to go back to a university sooner or later. One thing that is important, if this is your goal, is to be sure to continue to publish or otherwise participate in scholarship.

And, in the later stages of the job search, two important considerations: organizational structure (departments, units, schools, divisions, etc.) of the university or college in question, and the specific expectations they will have regarding teaching and service (teaching load, committees, summer programs, etc.).

There are many other great pieces of advice, but hopefully here I’ve featured a few that might be more important to job seekers and less commonly heard.

Reflections and Advice after Many Semesters

Last month, fellow blogger Brian Tholl wrote some advice about the graduate experience from his perspective as a first year graduate student. I think his advice is very useful and informative, and I want to reiterate some of it and add some advice from the perspective of a senior graduate student.

Brian is right, that transitioning into graduate studies can be difficult, and although the process is unique to each of us, much of his advice is likely to be useful to graduate students (even senior students like me). However, graduate school is itself a series of transitions, including admission, coursework, one or more qualifying exams, preliminary thesis work, intensive field work, lab work, research work or other studies, and thesis writing – not to mention teaching, personal life, and myriad other responsibilities unrelated the completion of the degree.

Graduate school is about transition — growth and change as a scholar, researcher, and educator is achieved throughout these transitions. It is important for graduate students to map out their graduate experience, which may vary from field to field, advisor to advisor, even student to student. But it is important to have reasonable expectations about what one wishes to accomplish at each stage of the degree program, information one should be seeking from one’s graduate director, advisor, and other faculty. Besides having day-to-day plans about when and where to invest one’s time, graduate students should have a structured and firm grasp of what their long-term goals are and how to accomplish them.

Organizing one’s free time is critical for first year students, and that will not change throughout one’s graduate studies and after. The responsibilities of graduate students (and in their futures as faculty, scholars, educators, or work) are usually task-oriented, rather than time-oriented. Significantly, these tasks are often very large, high-level goals, e.g. “write a thesis”, rather than simple short-term tasks, e.g. read XYZ paper in ABC journal for next week. Besides allocating time appropriately, it’s important to break down large tasks into small, feasible subitems and complete those subitems on a reasonable (not too lofty, not too lazy) schedule. Constructing these goals and setting them in a reasonable fashion allows students to complete seemingly impossible tasks, e.g. “write a thesis”, by working through a series of smaller, more tangible tasks.

Brian also mentions that graduate students should participate in social events, look after one’s health, and try to reduce stress. Learning to manage the demands and stresses of research work is a very important part of graduate school. It is indeed important for us to prioritize appropriately healthcare, healthy eating, stress-relief, and sleep. The time and money we spend on these may seem like a waste, because every waking hour could be spent working on our theses. But to the contrary, if we have a good perspective on our progress towards completing a degree, and taking care of our other obligations, e.g. teaching, we do have enough time.

Taking care of oneself will result in more effective and efficient research or teaching. There is a point of diminishing returns when a graduate student spends too much time working. I’m not saying don’t work hard — hard work is important, and we are all aware that graduate students may work 60+ hours per week, but in the remaining 80 or 100 hours, we should set aside time for eating right, sleeping adequately, and taking some personal time to relax and stay sane.

I will offer some other practical advice briefly, as a list rather than expounding at length:

  • Have conversations. All the time, with everyone, talk about research, talk about teaching, talk about anything. Use your friends and colleagues as sounding boards, discuss your challenges with your advisor, and really listen when people do the same with you.
  • Attend seminars, workshops, conferences. These are informative and fun, and should help you expand your scholarly boundaries. Even if the topics are tangential or unrelated to your research, you will learn much and may find hidden connections or new interests. This is also a useful way to do “networking.”
  • Teach your own classes whenever possible. This requires a huge time investment compared to TA work in recitations, lab sections, grading, etc. but is an incredibly important part of professional development. Even if you only teach one class for one summer, that’s a great opportunity to get teaching experience.
  • Don’t sweat the small stuff, at least, not every time. Graduate students aspire to be scholars, educators, and leaders in their fields of study, which often requires incredible attention to detail. However, it is important to recognize which details are crucial and which are expendable. There is that metaphor about the forest and the trees — don’t get lost!

Marie desJardins has written an excellent, lengthy guide to being a graduate student, and while not all her advice will apply to every single student, it is actually very relevant to most of us and is a very useful and frequently cited tome of advice for graduate students.

Conferences in Mathematics

Attending conferences is an important part of academic work. Conferences help us share our research with one another, find new collaborators and research topics, and keep up to date on our fields of interest.

I recently attended a bi-annual conference hosted by Integers (The Electronic Journal of Combinatorial Number Theory). I should say that my travel was generously supported by the conference organizers (i.e. the journal, via the NSF I believe) and my department & advisor, although I should say that one part of the conference experience is waiting with bated breath to get reimbursement forms processed. The government shutdown doesn’t help with that long wait either.

Rather than talk about the math, which isn’t really the point of this blog, I wanted to share some of the peripherals — the details of the conference, its format, what the experience is like. I have heard stories from other fields of study, and conferences seem to be very different from place to (figurative) place.

The departure is usually a bit of a rush of packing and preparing slides for presentations. Beamer (or equivalent) have become the de facto presentation method at math conferences, having (somewhat recently…) displaced the long-reigning overhead projector. After a day of travel, including a bit of a drive to Carrollton GA (home of UWG), I got some sleep before the first long day of presentations. I don’t travel much, and it is certainly stressful and tiring, but in the end I do enjoy it, especially driving.

Conference presentations are usually split into short (20 min) and long (50 min) talks, the later being given by specially designated (invited, plenary, keynote etc.) speakers of the conference. Most talks aim to communicate some new results, ideas, or insights into some type of research, and even for a specialized conference, there is a great deal of diversity in the subject matter. Some speakers speak to the general conference audience, while others speak to the very best experts in their slice of the research world. Many of the most interesting talks, to me at least, don’t probe into the depth of the subject, but give a gentle introduction or overview, and then outline or sketch the major new results or ideas. I’m more of a breadth-first guy.

The conference lasts for several days, as many conferences do, with talks back-to-back from about 9 to 5 every day. There are breaks for meals and coffee, and many conversations — professional and social — branch out from the main group during and after the sessions. Conferences are a great way to meet, re-meet, or quasi-meet people. I re-met Brian Hopkins, who has done some work related to my talk, and Bruce Landman, who has also worked in a related area (and is one of the conference organizers). Both of them (and several other audience members) had interesting questions and comments following my talk — one of the best parts of a conference is getting insightful feedback from colleagues. But I also met a few people more socially. I had a short chat about hockey with Cam Stewart after overhearing him talking about the sport, and sat at a table during the conference banquet with Steve Butler, Mel Nathanson, and Neil Hindman. Mel proposed an interesting problem at the conference that provided stimulating discussion and that I’ve found to be an interesting diversion even after the conference ended.

I also met other grad students like myself, many from closer to UWG (from schools like UGA, Georgia Tech, etc.), including Kate Thompson, whose advisor Jon Hanke  spoke here at Rutgers (by coincidence) only a few weeks after the conference (he was not at the conference). Making acquaintances can be quite beneficial — in this case, Kate and Jon know quite a bit about quadratic forms, which is something that is at least tangentially related to some long-term research ideas I’ve kicked around for a little while (but quadratic forms, on the whole, is a foreign subject to me). One day, if it comes up, I know somebody I can email if I stumble across questions or ideas I can’t wrap my head around.

Conferences in other fields can (apparently) be very different — my friends in the humanities tell me that conferences sometimes (often? always?) consist of reading papers aloud and asking prepared questions, while I have seen that some (many? most?) scientific conferences revolve around poster sessions and other such media. But for us in math, at least in my experience, it is a long sequence of presentations aimed (usually) at general information for the research-level audience, describing research ideas and perspectives and leaving technical details for the published page. I like this format, especially because it promotes dialog, discussion, and feedback — and helps people like me reach out a bit and meet others with similar interests and ideas in mathematics.

What I Learned This Summer

Summer teaching is a unique experience for many graduate students. For students in many disciplines, it may be the first, primary, or only chance to teach one’s own class. In addition to being an opportunity for graduate students to transition to instructor roles, summer courses also give students and instructors alike different opportunities than a Fall or Spring term.

For me, this summer presented the chance to develop new and exciting (to me, at least) materials for the course MATH 244, which is a course on differential equations geared mainly to engineers. Without changing the overall curriculum of the course, I decided to integrate computer-based exercises (small & large) to give students the chance not only to learn computer skills that accompany the mathematical material in the course, but to use computer-based work to aid them in learning the rest of the material.  The goal of these changes was to improve learning.  Although the course was not long enough to establish serious, long-term, in-depth skills, the experience should serve as a useful introduction to particular sorts of software and to computing skills.

an epidemiological model

I also believe the use of the computer modernizes some of the other instruction, and benefits students with algorithmic, visual, or kinetic learning styles. I believe the ability to manipulate mathematics and see significant visual output in real-time has a profound impact on how students understand concepts like the stability of equilibrium-solutions to differential equations.

I would also argue that this approach helped better organize the course, both logistically and in students’ minds, as components of the course related to computation and visualization were not segmented into awkward places throughout the term. By making computation and visualization more central, and more hands-on, students more easily integrated this material with the theoretical and non-computational methods.

an information flow model

This also effectively separated the computational and visual elements of the course from exams, where asking students to perform tasks better suited for a computer limits the assessment. Students were instead asked to delve into complex computer-based tasks over a longer period of time, as “projects”. (This, as a side-effect, provided some relief from high-stakes testing.)

Students used complex computational and graphical methods.

Student evaluations seemed to indicate that students perceived this part of the course as beneficial: Instructional survey scores for “I learned a great deal in this course,” “I rate the teaching effectiveness of the instructor as,” and “I rate the overall quality of the course as” were substantially higher than average. Individual comments reflected positively on the use of computers for assignments and for in-class demonstrations. Five respondents to the instructional survey (over 25% of the class) indicated that computer-based work was how “this course or the instructor encouraged [their] intellectual growth and progress.”

A population dynamics model

Of course, this was challenging for students, but despite the fast-paced nature of a summer course, most students did not feel overloaded with work. For every student who believed there was “extra work” due to the additional computer-based workload, there was another who realized that this was eliminating a significant amount of alternative work to be done without computer to cover the same material.

Here is an interactive demonstration similar to those used and produced by students. The Wolfram CDF Plugin is required.

Images used in this entry are used under fair-use guidelines. They are excerpts from student-generated work in the course described in this blog post.

Intergalactic Planetary Research is Useful Too

Fellow blogger Michael posted this entry recently, and rather than just write a comment, I thought I would chime in on this issue as well. I won’t try to summarize completely his relatively short blog entry, but to put it briefly, Michael reminds scientists that to solicit funding from the public or public policy-makers, scientists must engage the public and inform them of the content and benefit of their scientific work.

I think a fair amount of what Michael said applies across-the-board to any variety of scholarship, and on the whole, I very much agree with his point and support it. However, I would like to pose a question. Scientific work may suffer a lack of funding due to waning interest, familiarity, or other such motivation on the part of the public and/or policy-makers, and I would agree that there is some onus on scientists to reinvest in relations with said parties. But is it not also true that statements such as “I personally don’t see the benefit, or “understand” or “like or “appreciate” this research (or science as a whole)” are not an excuse to fail to support the endeavors of scientists that work toward the public good, be it through basic or applied science, research or education?

I agree that public relations, outreach, etc. are very important, and raising awareness of the importance of science (or any other type of scholarship) would very much help bring back much-needed public support to the policy debate regarding funding for research. I suspect improving public education would do likewise. But I don’t think efforts to gain public support should hinge on whether the public is properly educated about a specific scientific endeavor, nor on whether this endeavor has an immediate & direct impact on the public good (e.g. climate change or healthcare). Basic research science and mathematics has sometimes been described as a money-pit into which we dump millions of dollars and get no “products” or “solutions” because researching bugs or quasars or quarks or Lie groups seems to be useless. This belies the fact that the applied sciences, as well as most fields of engineering, technology, communications, etc., rely heavily on the existing and expanding body of research in basic science and mathematics.

NASA is one example of a publicly-funded institution that supports not only scientific research but also its own space-exploration program. It has been a leader in the scholarship of astronomy, engineering of many types, and scientific leadership. Now its funding has been cut because policy-makers (and perhaps the public) think space exploration is not important.  And there are many reasons this is the case, among which I do count Michael’s important and very agreeable point. One role of scientists (or any researcher, publicly funded or not) is to communicate effectively the nature, role, and importance of his/her work to the general public. However, I am arguing that this should not extend so far as to require researchers to educate the public on the entirety of science, as this is impractical and infeasible.

Research is a long-term endeavor that navigates twists and turns, hinges on unknowns, and takes long spans of time. It also requires us to accept that projects may fail to produce good results, or that the results may not lead immediately to new solutions to applied problems. The same is true of funding scholarship and research — not every researcher will be successful as an “investment” in the short term, and some may leave research altogether, but we do not subject every first-year graduate student to an inquisition to determine if they will solve a world-changing real-life problem in 10 years and only fund those who demonstrate this. In aggregate, it is important to fund research (and researchers) sufficiently well without demanding guarantees of success, or an accounting of immediate gains from this investment.

Organizing Events and Programs

Organizing an event can be incredibly taxing and difficult, especially for a graduate student. However, grad students are often brought into projects of this sort. It provides an event or program with capable staff or assistants of whatever sort, and also provides the student with an important type of experience. The managerial and administrative skills grad students can learn and refine from these experiences is important and useful. If anything can teach time-management, putting together a conference or workshop certainly can.

The type of work undertaken can be varied, as can be the time-commitment and intensity of work. Some students may help with the logistics of a conference and wind up incredibly busy for a 3-day period, while others may be junior members of the organizing committee and wind up working a moderate amount* over a longer period of time.

*And keep in mind, such duties and such work are undertaken in addition to existing obligations towards research, teaching, or coursework. So “moderate” is more than it sounds, perhaps.

From my perspective, these skills are sometimes hard to describe or quantify. Some of the skills may be specific to the type of event being organized, while others may be widely applicable. Having been graduate coordinator for the DIMACS REU for several years, I believe some of the experience may only be applicable in scenarios with undergraduate research. Other skills, however, may transfer to scenarios like organizing a conference or working within a department or university bureaucracy. In some sense, one learns how to do things, how to get things done, whether that means learning to adapt to certain scenarios, understanding how to navigate certain structures, or simply having the experience of making something happen. In the future, stepping up to the figurative plate will be easier and more natural.

One important virtue in organizing events and programs that I have come to value as almost universally applicable and of great importance is this: Set yourself up to succeed. Front-load the work, make sure it is done right, have a plan, and always be as prepared as possible.  Don’t forget to follow up on important emails. Make sure that contingencies have been covered. Accidents will happen, disasters will occur, and you will make mistakes. Have a timetable, have back-up plans, and so on.

That sounds like many principles, but to me it really is one coherent guiding idea. Success in organizational and administrative tasks can be decided, or at least heavily weighted, by the organizational and managerial efforts invested, especially those invested early. That lesson, and a little experience, can help a capable grad student or young faculty member successfully bring together virtually any meeting, conference, project, or program.

Around Town

In New Brunswick (and the surrounding area), there are many opportunities to relax, have fun, and get out of the house/office. There are a variety of ways to enjoy the diverse offerings of New Jersey of all types: cultural, recreational, gastronomical, natural, musical, and more. I’ll list just a few of my personal favorites, in no particular order and with no attempt to fairly represent all of the possible local attractions and activities.

The State Theater of New Jersey

The State Theater is especially good for students because Rutgers-affiliated discounts make many of the shows accessible for a good price (and not just in the very back row either). I have enjoyed music and dance performances here, as well as a few comedy routines and at least one musical. There are many different types of acts coming to the State Theater at any given time, so it’s worth checking their schedule periodically for your interests. The theater district of New Brunswick also offers the Cabaret Theatre, the George Street Playhouse, and the Crossroads Theatre Company.

Image source: [c]

The Court Tavern

New Brunswick has had a vibrant local music scene for decades, including many pioneering and influential bands. Although recently closed for a few months, the Court Tavern has reopened recently and continues to provide a venue for local musicians. Nearby places to eat, including Hansel ‘n’ Griddle and Destination Dogs, serve up food suitable for before the show (or between bands).

Image source: [c]

New Brunswick has many bars, pubs, and other venues in that spectrum of eating & drinking, but only Harvest Moon offers beer brewed right on-site, along with its classic fare of dinner options. The vegetarian chili is a hidden gem on the menu, which includes many options that are easy to enjoy. Be sure to try one of their many varieties of beer (they won’t sell you anybody else’s!), and if you like it, take some home in a growler.

There are many other similar restaurants in New Brunswick, including Tumulty’s and The George Street Ale House, both of which offer good beer and nice pub food. Tumulty’s is a nice alternative for a more laid-back and traditional pub experience, and GSAH has a wide selection of beer as well as a more pricey menu of “gastro-pub” fare.

Image source: [c]

Stelton Lanes Bowling Alley

I enjoy spending time with friends bowling. Knocking things over is fun (sometimes). In addition to the enjoyment of time spent with friends, bowling is an activity easy to enjoy even for beginners, but it’s not boring or highly competitive (unless you have really intense friends). You’ll be renting shoes, most likely, so be sure to wear socks! Another nearby bowling alley is the Brunswick Zone.

Image source: [c]

The Edison Diner

New Jersey is known for its diners, often open late (or 24-7) and offering a variety of classic meals, snacks, and beverages. I am a somewhat regular customer at the Edison diner. For those days when there’s no time to cook or it’s too late to do otherwise, this diner is open 24-7 and even has free wifi. Outside peak hours, the diner is sometimes (but not always) calm and relaxed and can be a nice place to get some peace and quiet some days.

A mention should go to another favorite diner of mine, The Skylark Diner. This diner is a bit further away and is not open 24-7, but it offers a more creative/non-traditional menu and a fair selection of alcoholic beverages. (It is thus a bit more loud and crowded, and the prices are a bit higher.)

Image source: [c] You can visit the Edison Diner’s website by clicking this link, but please beware, it is a flash-only website that makes a substantial amount of noise: The Edison Diner.

Middlesex County Parks

It may be freezing out and dark at 5pm during winter, but even in winter it’s nice to spend time in one of the many parks in the area. Donaldson Park in the boro of Highland Park offers many facilities, including walking paths, basketball courts, exercise stations (pull ups, monkey bars, etc.), dog pens, and more. The park is prone to flooding, which we have experienced a few times in recent history, but it is generally scenic and enjoyable. Nearby Johnson Park has similar facilities as well as a sanctuary for abandoned animals rescued by the county.

Image source: [pd]

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.

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.

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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.

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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.