Against the “Research vs. Olympiads” Mantra

Posted on by Evan Chen (陳誼廷)

There’s a Mantra that you often hear in math contest discussions: “math olympiads are very different from math research”. (For known instances, see O’Neil, Tao, and more. More neutral stances: Monks, Xu.)

It’s true. And I wish people would stop saying it.

Every time I’ve heard the Mantra, it set off a little red siren in my head: something felt wrong. And I could never figure out quite why until last July. There was some (silly) forum discussion about how Allen Liu had done extraordinarily on math contests over the past year. Then someone says:

A: Darn, what math problem can he not do?!

B: I’ll go out on a limb and say that the answer to this is “most of the problems worth asking.” We’ll see where this stands in two years, at which point the answer will almost certainly change, but research \neq Olympiads.

Then it hit me.

Ping-pong vs. Tennis

Let’s try the following thought experiment. Consider a world-class ping-pong player, call her Sarah. She has a fan-base talking about her pr0 ping-pong skills. Then someone comes along as says:

Well, table tennis isn’t the same as tennis.

To which I and everyone else reasonable would say, “uh, so what?”. It’s true, but totally irrelevant; ping-pong and tennis are just not related. Maybe Sarah will be better than average at tennis, but there’s no reason to expect her to be world-class in that too.

And yet we say exactly the same thing for olympiads versus research. Someone wins the IMO, out pops the Mantra. Even if the Mantra is true when taken literally, it’s implicitly sending the message there’s something wrong with being good at contests and not good at research.

So now I ask: just what is wrong with that? To answer this question, I first need to answer: “what is math?”.

There’s been a trick played with this debate, and you can’t see it unless you taboo the word “math”. The word “math” can refer to a bunch of things, like:

  • Training for contest problems like USAMO/IMO, or
  • Learning undergraduate/graduate materials like algebra and analysis, or
  • Working on open problems and conjectures (“research”).

So here’s the trick. The research community managed to claim the name “math”, leaving only “math contests” for the olympiad community. Now the sentence

“Math contests should be relevant to math”

seems totally innocuous. But taboo the world “math”, and you get

“Olympiads should be relevant to research”

and then you notice something’s wrong. In other words, since “math” is a substring of “math contests”, it suddenly seems like the olympiads are subordinate to research. All because of an accident in naming.

Since when? Everyone agrees that olympiads and research are different things, but it does not then follow that “olympiads are useless”. Even if ping-pong is called “table tennis”, that doesn’t mean the top ping-pong players are somehow inferior to top tennis players. (And the scary thing is that in a world without the name “ping-pong”, I can imagine some people actually thinking so.)

I think for many students, olympiads do a lot of good, independent of any value to future math research. Math olympiads give high school students something interesting to work on, and even the training process for a contest such that the IMO carries valuable life lessons: it teaches you how to work hard even in the face of possible failure, and what it’s like to be competitive at an international level (i.e. what it’s like to become really good at something after years of hard work). The peer group that math contests give is also wonderful, and quite similar to the kind of people you’d meet at a top-tier university (and in some cases, they’re more or less the same people). And the problem solving ability you gain from math contests is indisputably helpful elsewhere in life. Consequently, I’m well on record as saying the biggest benefits of math contests have nothing to do with math.

There are also more mundane (but valid) reasons (they help get students out of the classroom, and other standard blurbs about STEM and so on). And as a matter of taste I also think contest problems are interesting and beautiful in their own right. You could even try to make more direct comparisons (for example, I’d guess the average arXiv paper in algebraic geometry gets less attention than the average IMO geometry problem), but that’s a point for another blog post entirely.

The Right and Virtuous Path

Which now leads me to what I think is a culture issue.

MOP alumni prior to maybe 2010 or so were classified into two groups. They would either go on to math research, which was somehow seen as the “right and virtuous path“, or they would defect to software/finance/applied math/etc. Somehow there is always this implicit, unspoken message that the smart MOPpers do math research and the dumb MOPpers drop out.

I’ll tell you how I realized why I didn’t like the Mantra: it’s because the only time I hear the Mantra is when someone is belittling olympiad medalists.

The Mantra says that the USA winning the IMO is no big deal. The Mantra says Allen Liu isn’t part of the “smart club” until he succeeds in research too. The Mantra says that the countless time and energy put into running each year’s MOP are a waste of time. The Mantra says that the students who eventually drop out of math research are “not actually good at math” and “just good at taking tests”. The Mantra even tells outsiders that they, too, can be great researchers, because olympiads are useless anyways.

The Mantra is math research’s recruiting slogan.

And I think this is harmful. The purpose of olympiads was never to produce more math researchers. If it’s really the case that olympiads and research are totally different, then we should expect relatively few olympiad students to go into research; yet in practice, a lot of them do. I think one could make a case that a lot of the past olympiad students are going into math research without realizing that they’re getting into something totally unrelated, just because the sign at the door said “math”. One could also make a case that it’s very harmful for those that don’t do research, or try research and then decide they don’t like it: suddenly these students don’t think they’re “good at math” any more, they’re not smart enough be a mathematician, etc.

But we need this kind of problem-solving skill and talent too much for it to all be spent on computing R(6,6). Richard Rusczyk’s take from Math Prize for Girls 2014 is:

When people ask me, am I disappointed when my students don’t go off and be mathematicians, my answer is I’d be very disappointed if they all did. We need people who can think about these complex problems and solve really hard problems they haven’t seen before everywhere. It’s not just in math, it’s not just in the sciences, it’s not just in medicine — I mean, what we’d give to get some of them in Congress!

Academia is a fine career, but there’s tons of other options out there: the research community may denounce those who switch out as failures, but I’m sure society will take them with open arms.

To close, I really like this (sarcastic) comment from Steven Karp (near bottom):

Contest math is inaccessible to over 90% of people as it is, and then we’re supposed to tell those that get it that even that isn’t real math? While we’re at it, let’s tell Vi Hart to stop making videos because they don’t accurately represent math research.

Addendums (response to comments)

Thanks first of all for the many long and thoughtful comments from everyone (both here, on Facebook, in private, and so on). It’s given me a lot to think about.

Here’s my responses to some of the points that were raised, which is necessarily incomplete because of the volume of discussion.

  1. To start off, it was suggested I should explicitly clarify: I do not mean to imply that people who didn’t do well on contests cannot do well in math research. So let me say that now.

  2. My favorite comment that I got was that in fact this whole post pattern matches with bravery debates.

    On one hand you have lots of olympiad students who actually FEEL BAD about winning medals because they “weren’t doing real math”. But on the other hand there are students whose parents tell them to not pursue math as a major or career because of low contest scores. These students (and their parents) would benefit a lot from the Mantra; so I concede that there are indeed good use cases of the Mantra (such as those that Anonymous Chicken, betaveros describe below) and in particular the Mantra is not intrinsically bad.

    Which of these use is the “common use” probably depends on which tribes you are part of (guess which one I see more?). It’s interesting in that in this case, the two sides actually agree on the basic fact (that contests and research are not so correlated).

  3. Some people point out that research is a career while contests aren’t. I am not convinced by this; I don’t think “is a career” is a good metric for measuring value to society, and can think of several examples of actual jobs that I think really should not exist (not saying any names). In addition, I think that if the general public understood what mathematicians actually do for a career, they just might be a little less willing to pay us.

    I think there’s an interesting discussion about whether contests / research are “valuable” or not, but I don’t think the answer is one-sided; this would warrant a whole different debate (and would derail the entire post if I tried to address it).

  4. Some people point out that training for olympiads yields diminishing returns (e.g. learning Muirhead and Schur is probably not useful for anything else). I guess this is true, but isn’t it true of almost anything? Maybe the point is supposed to be “olympiads aren’t everything”, which is agreeable (see below).

  5. The other favorite comment I got was from Another Chicken, who points out below that the olympiad tribe itself is elitist: they tend to wall themselves off from outsiders (I certainly do this), and undervalue anything that isn’t hard technical problems.

    I concede these are real problems with the olympiad community. Again, this could be a whole different blog post.

    But I think this comment missed the point of this post. It is probably fine (albeit patronizing) to encourage olympiad students to expand; but I have a big problem with framing it as “spend time on not-contests because research“. That’s the real issue with the Mantra: it is often used as a recruitment slogan, telling students that research is the next true test after the IMO has been conquered.

    Changing the Golden Metric from olympiads to research seems to just make the world more egotistic than it already is.

Published by Evan Chen (陳誼廷)

I am a math olympiad coach and a PhD student at MIT.

33 thoughts on “Against the “Research vs. Olympiads” Mantra”

      2. For some reason I can’t reply to your most recent comment, so I’ll just reply here.

        I realized that the final paragraph of my previous reply was ambiguous. I was in no way criticizing Jacob Barnett, and when I said “immature young kids,” I was referring to some of the AoPS members on that thread. I hope this clears my comment up a bit.

        I’m also of the belief that doing research without the necessary qualifications can be a meaningful experience. The problem seems to be that people who don’t have much knowledge of some field also generally have little knowledge about the breadth of the field, which can lead to some extremely unrealistic expectations. If less experienced students applied for research opportunities with some more realistic expectations, they could learn something valuable. For instance, one of the big problems is that academia has very little public outreach. It’s almost like a cult. Maybe there is some Hollywood notion of the reclusive mathematician who sits in front of the blackboard all day and produces some equations out of thin air, but the general public doesn’t know anything about what any of it is actually like. They don’t know how collaborative it all is. They don’t know that there is a lot of digging to be done, a lot of relevant literature to catch up on, a lot of references to consult. Nobody really comes up with magical insights out of nowhere that revolutionize the entire field. So to make up for this lack of exposure, some ambitious kids could see the process up close, which can help them decide earlier on if they even want to pursue research. But the earlier point of diminishing returns is still true here. At that stage, there is far more to gain by learning about the general process than there is by trying to achieve mastery of the domain.

        And I happen to think differently on the delusion issue: I think for most inexperienced students, getting them involved in research they are not prepared for is more likely to disillusion them, rather than delude them.

        Oops, I feel like I’m kind of getting off topic.

        Like

    2. You some good arguments. I mostly agree with you about that calculus trap article. While the intentions of that article are good, the actual point may not be presented explicitly enough for less mature students to see right away, which can be somewhat disastrous. I find myself constantly having to tell younger students that the article doesn’t actually recommend that you put off calculus and higher math for as long as possible; that calculus was only used in the analogy because it’s usually perceived by students and the general public as being the “last” or “highest” high school math class. In fact, judging by the response the article got on Reddit, even many of the older students find themselves confused about the actual meaning.

      And I agree with your insights about the diminishing returns you get when you spend an excessive amount of time on olympiad math. I think this is true for anything, really. If the activity is done with the intent of getting indirectly good at something, but not necessarily the activity itself, then people should be cautious about how much time they’re spending on it.

      But about that Jacob Barnett thing… It seems like most of the hate was mostly a knee-jerk reaction towards the excessive media sensationalism. Claiming that a 12-year-old child “expanded Einstein’s theory of relativity” and then focusing entirely on him computing some trivial integrals is certain to leave some people skeptical. And being immature young kids, some will make some pretty strong claims without knowing much about what they’re talking about. I’m not very convinced that it has much to do with the calculus trap.

      Like

    3. > I don’t think that the articles in the links you provided in the beginning of this article were reciting the “mantra” in order to “belittle Olympiad medalists” – I think the purpose of those articles was to encourage students who did not perform well in Olympiads to not think they are “bad at math”.

      Sure, that’s certainly valid. Someone (AM) pointed out to me there’s to some extent a “bravery debate” dynamic here, which I think is correct. (http://slatestarcodex.com/2013/06/09/all-debates-are-bravery-debates/)

      Like

  3. To me, the most important qualitative distinction between contests and research has always been this: in research math, you’re supposed to come up with original work, to push the boundary of human knowledge if you want to be a bit more poetic; but in contests, you’re solving “cut-and-dried” problems usually chosen for having neat, approachable, known solutions — problems others have already solved for you, so you’re not creating the same sort of value.

    Possibly less value than world-class ping-pong and tennis players, I might argue. I don’t think you can make a career out of solving olympiad math problems (do tell me if you know a way otherwise), but you can out of playing ping-pong and tennis, because the latter are better spectator sports; people enjoy watching them that much more, so by playing you’re creating value for them.

    (although certainly you can make a career out of teaching people math or even just grading solutions they wrote, which might involve solving olympiad problems, but the value you’re providing there is pedagogical, not exactly what these contests test you on or prepare you for. Similarly, I am generally more impressed by people writing math contests than doing well at them!)

    I am still arguing with myself over the rest of the post and other points brought up in the comments. Perhaps more later.

    Like

    1. “Creating value” is completely subjective. By doing pure math research, how much value are you really creating? Your work might not ever be applied to anything in the real world, and more than likely the only people who will ever see your work are other mathematicians. Yet, we can argue that it does have value, because developing the field with the intent of expanding the total body of knowledge is sure to indirectly lead to important breakthroughs that do have practical applications. We can argue that math is valuable and beautiful in its own right to a particular audience, and that the practical applications only come as side benefits. And we can even argue that the intellectual culture fostered by pursuing knowledge for the sake of knowledge will indirectly lead to a better world.

      It’s just as easy to come up with similar arguments in favor of being a competitor. Without competitors, contests would not exist. And Evan has already covered many of the benefits of math contests, so I’ll spare you the repetition.

      But aside from the utilitarian notions of “value,” why do people feel the need to step in and belittle competitors with their rude olympiad-to-research comparisons whenever other people want to show that competitor some due respect? Where do people get the idea that making original contributions is the only intellectually respectable accomplishment, and that everything that is not directly related deserves relentless criticism?

      Like

      1. B’s comment in the post was rude and uncalled-for. Allen Liu and others who do well at math contests have accomplished something great and deserve a lot of respect; they’re pretty likely to accomplish more in the future. Math contests can help you achieve personal growth and meet a peer group and learn general problem-solving skills.

        I agree with all that, but still disagree that “‘Creating value’ is completely subjective. I should just have avoided such a vague term as “value” in the first place, so, with no subjective words: Many people are paid full-time to do math research. Nobody is paid full-time to solve contest problems with known solutions under time restrictions.

        Based on this, I would say society has come to the consensus that the former career creates more value than the latter, perhaps based on the reasons you gave.

        (Research is definitely not the only such more-valued-by-society career — I agree with Evan when he points that there are many other careers where we should be encouraging good problem-solvers to go, and it’s harmful to paint research as the “right and virtuous” career and other careers as somehow lesser. But I find the jump to “the Mantra belittles olympiad medalists” a non sequitur. Problem-solving in other careers does not rely on solving contest problems with known solutions under tight time constraints either.)

        And I think this is worth pointing out because I think math contests give diminishing returns after a certain achievement level, at which point you can do other things that will advance your long-term career more (which is not to say continuing to do contests is bad, if you enjoy it and want to); but there is a certain tendency for contestants to keep doing contests for longer just because it’s familiar and they’ve trained so long and invested so much effort for that specific purpose. I would say the takeaway from the Mantra for medalists should be, “Congratulations on medaling, but you won’t be able to rest on your laurels and only keep practicing the exact same skills for being good at olympiads forever.”

        If a gold medalist feels that this belittles them or their achievement, that’s unfortunate, and I can’t speak for all the authors of the linked articles, but it’s not my intention.

        Taking a step back, though, I also want to reiterate an important point from Anonymous Chicken’s first comment:

        the purpose of those articles was to encourage students who did not perform well in Olympiads to not think they are “bad at math”.

        Feel free to criticize the articles for the message they send to medalists, but don’t forget that far more people who read them will never have gotten anywhere near an olympiad medal. I think those people could use a few encouraging words, especially if they like math but find math contests discouraging.

        Liked by 1 person

    2. I think there’s a whole debate that you could do about “value” or whatever. Agree it’s an interesting question worth thinking about (and I don’t think the answer is so obvious; undecided myself).

      Also happy to hear I’m not the only one that values the pedagogy component of contests :)

      Like

  4. I agree with most of your stuff, but I disagree with almost every word of this post.

    The problem is tribal. You’re reacting strongly to an attack on your tribe while being totally oblivious to the (much stronger, more numerous, and more harmful) attacks that it’s firing back. Several harmful ‘mantras’ that the Olympiad tribe use are:

    1) Only Olympiad people are ‘legit’. The truest measure of intelligence is performance on Olympiads.

    People with research experience, like Jacob Barnett, are viewed with suspicion. The general idea is that, even if they know a lot of stuff in some narrow subfield, they _must_ think slower than the Olympiad gods, and hence they’re less legit.

    Having now met a lot of people who were too busy learning advanced stuff in high school to do Olympiads, I can say confidently that this is absolutely not the case. But when people read Terence Tao’s biography, they only care about his Olympiad performance, even though everything that came after is much more important.

    2) The only thing that matters is ‘being legit’. Smart people (i.e. Olympiad people according to (1)) are the only people worth talking to or working with.

    Olympiad people wall themselves off, in places like SPARC (>80% Olympiad when I went), Random, and 3W. This causes some serious myopia and destroys the horizon-expanding benefits of college. How can you speak about economic policy, for example, if you don’t know anybody who had to work during high school? If you don’t know anybody that worries about being able to find a job? Not everybody can or will work in Silicon Valley and Wall Street.

    I also ran into this several times when checking out finance companies. The key benefit, repeated over and over again, was that you got to work with Smart People. That your job was to rake in the money you deserved, from your ability to outsmart the market. Needless to say, mantra (2) is an extremely useful recruiting tool for finance, which sucks in Olympiad people like crazy. And though I agree with you on the questionable impact of pure math research, finance is definitely worse.

    3) The only useful application of mental energy is solving hard technical problems.

    Communication, teaching, and all creative and physical activities are severely undervalued. This is part of what the ‘research vs. olympiads’ mantra is trying to get at: that to be a well-rounded mathematician, you have to do much more than crunch out lemmas quickly.

    Worse, mantra (3) pushes Olympiad people out of many fields where they could make substantial impact. Pretty much any job that involves directly helping human beings is ruled out under (3). Public service is out. Law, medicine, and engineering is out. Administrative roles are out, despite their enormous possible impact. Software engineering is almost ruled out, unless you’re at some fancy startup with only Smart People (as per mantra (2)).

    All in all, (1), (2), and (3) are collectively more harmful than the ‘olympiads vs. research’ mantra. I don’t see the latter as anything more than a reasonable response to those three.

    Liked by 1 person

    2. Another Chicken,

      I agree with you on basically all counts, but diverge from you at the conclusion. The Mantra isn’t saying “olympiads aren’t everything”, it’s saying that “olympiads aren’t everything *because research*”. It’s this last part that I object to.

      Changing the Golden Metric from olympiads to research doesn’t seem to be much of an improvement.

      Like

  6. Just one last point, I guess:

    I can very much sympathize with thinking there are jobs that shouldn’t exist because they don’t provide value-as-I-perceive-it to society. Exactly how much society should value any given career is, I agree, a whole other debate. But my objection can be phrased in purely pragmatic terms. If you keep doing something you value but society doesn’t currently, you won’t have a job and won’t be able to provide for yourself.

    Like

  7. I think I can summarize this for some people who might have missed the point. The point isn’t that Olympiads are perfect and exempt from any kind of criticism, or that they’re superior to any particular activity or use of time. The point is that out of all the valid criticisms, “Olympiads aren’t research” isn’t one of them. They’re two completely different things with completely different goals.

    I think any reasonable person would agree that Olympiads have some serious problems, but failing to emulate research is not one of them.

    Liked by 1 person

  8. Hi, I made the Allen Liu thread, and I wanted to make some comments here because I don’t feel good about how the discussion in that thread unfolded. I agree with you fundamentally about the attitudes toward contests vs. research, which is why I wish I’d been more deliberate about stopping the thread from going in that direction. The whole idea was my reaction of “people on a math forum should know about this achievement because they are the kind of people who would be able to appreciate the magnitude of it the most” and I’m a little disappointed that things went in a totally different direction.

    With that in mind, I think there’s also something to be said about the age demographic of AoPS in general; I think a lot of popular threads on the forum sort of degenerate into spam because many users are middle schoolers who do not necessarily have the maturity to navigate the intricacies of potentially a very complex discussion without being rude or overgeneralizing, etc.

    I think there is a component to this that should be mentioned, which is that the attitude about wondering how someone who does well on contests would do at research does not necessarily at heart have to have anything to do with the problem that you described. People do analogues of this all of the time in places outside of math, and can do it respectfully too, being just as enthusiastic about the achievement. Unfortunately, this is a very slippery slope, because evidently many people will jump on the “research is more interesting and important than contests” wagon as soon as this sort of thing is even mentioned, so we can’t have a fun (albeit useless, just as wondering if your favorite sports team or player will win the championship next season is) discussion without tons of people disrespecting other people, devaluing the field of contest math, and so on. It doesn’t help when some of the most influential mathematicians with the most far-reaching blogs (like Tao) give the impression that competitors are supposed to go into math, even if they aren’t meaning to and it’s just a lack of nuance in the description.

    The best thing that we can all do is to encourage people to pursue what activity they want as long as it isn’t harmful, and not tell them that one activity is superior to another activity. I think as long as we convince people from a young age not to let other people decide what’s important or worth doing for them, and to ignore the noise and elitism and bigotry that might surround them, we’re improving the experiences of others.

    Like

  9. I am entering high school and I am also an AOPS user.

    I do not plan to become a research mathematician. I am thinking about Silicon Valley tech startups or hedge fund quantitative trading as future career directions.

    I do contest math with two main objectives: 1) to pick up math knowledge/problem solving/intellectual thinking abilities that are needed for my desired careers choices, 2) to stand out on my college application. There are some other minor benefits I also reap (e.g. meeting smart people, contest problems and tournaments themselves being somewhat interesting — ARML, HMMT and so forth remind me much of going to an AAU basket tournament in another state with my “bros”).

    I allocate my hours to contest math training to target for USAMO qualification (but not MOP qualification) as I need to spend time on other things like coding, sports, debate, reading/writing, student government and potentially Intel tracked research projects.

    My question: Is my target of USAMO qualification optimal for my two main objectives? I think it is probably an overkill for objective 1), but not quite enough for objective 2). But maybe it is the best compromise?

    Like

  10. 1. The semi-dismissive attitude “let’s see how they do in the big leagues” is appropriate to the degree that the competition is the youth version of a professional activity. It makes sense as something to say about a junior world champion in chess or tennis, where nearly all of the stars go on to become professionals, but less so for the world championship of Little League Baseball where it is understood that there is not much relationship between being on a winning team and becoming a pro. The math olympiads are intermediate between those examples. Almost everyone gets some type of math-based university degree, about half try their hand at PhDs, maybe a third or a quarter actually become researchers. Those are high numbers compared to the retention rate from the high school elite to the pros in most sports.

    2. The same question is raised more intensively about Intel, Westinghouse etc since historically these have been pretty inefficient at identifying talent, and the olympiads have been better at that (but getting worse as they professionalize).

    3. Math olympiads are derivative of mathematics academia in almost every important way: the set of people who invent the problems, the origin of most (good) problems in some sort of research, the motivation for organizing and funding it (to encourage people who will research or apply non-competition mathematics in their careers). Every major national olympiad is run by people with mathematics doctorates.

    4. Although different from research in various ways that are often remarked upon, at the end of the day the olympiads are set up to identify and encourage the same sort of talent that is associated with top mathematics research, and (except for a few competitions watched or sponsored by industry) not really meant as a resume signal for elite jobs. I don’t think people really view Tao, Scholze or Perelman’s olympiad performance as a separate activity from their later careers, indeed, those were key steps in their progress toward becoming researchers. When their olympiad performance is held up as exemplary it is to validate the idea of olympiads as a science talent search. And this view has some objective justification.

    (This is in reply to the blog post, not the comments per se.)

    Liked by 1 person

  11. Also Evan Chen was just an example. Evan, I don’t know you personally and was not trying to tell you what career to follow – that is entirely up to you :P

    Like

    1. Also if you know a person like Jacob Barnett, do all you can to encourage them. I suppose what I meant by “Where I’m From” being hard to implement is that the systematic “production” of Jacob Barnetts by teachers seems more difficult and setting up programs to train contest math – but let me know if you figure out a way.

      Liked by 1 person

    2. In the below post, by “systematically produce Jacob Barnetts” I mean allow more people to give stuff like that a try in a variety of different fields and see how they like it, at a young age.

      Like

  12. A witty response to the mantra: Pure mathematicians do math for its own sake, independent of any value to applications. Can’t contest math people do contest math for its own sake independent of any value to pedagogical applications including research?

    Like

  16. Math competition is helping students to improve their skill. In my opinion, they should start taking it as early as they can in order to build a firm foundation to handle higher level math. My son is taking math exercises and math competition via Beestar, which is an online platform that provides different levels of math materials.

    Like

Leave a Reply to Evan Chen (陳誼廷) Cancel reply

Fill in your details below or click an icon to log in:

Gravatar
WordPress.com Logo

You are commenting using your WordPress.com account. ( Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. ( Log Out /  Change )

Cancel

Connecting to %s