Math Education & Competitions

[Mehmet Kayaalp]

Dear All:

I am uneasy and ambivalent about the role and importance of math competitions and want to hear from you what you think about them.

In order to be "competitive" in garden variety math competitions (i.e., excluding very top notch ones such as Math Olympiads), the student has to have practiced on the same topic so repetitively so that s/he would not need to spend much time about such questions in the competition, quickly recognize the patterns of interest in those questions, run the algorithms in his/her mind rapidly and accurately, and move to the next question without any hangover from the previous ones.

The benefit of this approach is, obviously, being "fluent" in basics. When fluent in math, one can learn new math concepts easily, prosper in their undergrad / grad math / engineering studies, and would not be bogged down by the basic math when solving engineering problems in their workplace.

I believe there is also a downside of this approach. Practicing basics to that extent may mean focusing on a well-circumscribed, fixed (or even rigid) curriculum with less depth and breadth and little exploration, and reducing math tests into a long but quick check list.

The habits and attitudes that we want our kids to gain are not this. We want our kids to focus on the math problem at hand even when the solution is not obvious, and try various approaches to develop their solution. 

The tricky part at our end is to provide the right question to the student at the right time. Attention span is one of the few operative words in this context, which improves as the kid grows. But attention span is not a fixed entity, either. The amount of time that a kid may spend on a math question with no light at the end of the tunnel is far different than the time that s/he could spend on a video game while s/he is in the flow. The difference can be huge --- frustration at one end, pure bliss at the other.

The answer may be moderation and balance in repetition (i.e., competitions), breadth (i.e., glimpses on new math topics), depth (i.e., analyzing more complex facets of the known concept) and exploration (i.e., solving puzzles and math circle like activities).If so, how can we gauge the student in order to achieve the right balance? 

Where is the place of math competitions in this balance? Should competitions be part of math education? Does your answer depend on the student's type or character?

Thanks,

--mehmet

PS. Another operative word probably is grit. How should we develop grit without frustration? When a math question looks to you as an intriguing puzzle that you cannot set aside and cannot resist but think about it all the time, you find yourself in the flow. Being in such a mental state frequently may help you to do it again even if the question at hand seems intractable and uninteresting.

Perhaps we should develop such intriguing, irresistible puzzles that put the student into the flow. Are there any commonalities in those questions? Or are those characteristics very much student dependent? Or perhaps both? If there are distinguishing attributes in those puzzling questions and if we can figure them out, then we might be able to build a compendium. 

I suspect such questions should look solvable but as you attempt to reach it, you could discover that the solution is rather elusive. "Look solvable" means that the student should be ready to attack the question with the right tool set in hand. So, it depends on the readiness of the student. It should also "deceive" the student with a false impression of being easy. But at the same time, it should not be too hard, either. Such a question might need to combine two or more distinct and distant math concepts. 

[Maria Droujkova]

On Thu, Mar 26, 2015 at 3:44 PM, Mehmet Kayaalp <mehmet....@gmail.com> wrote:

Dear All:

I am uneasy and ambivalent about the role and importance of math competitions and want to hear from you what you think about them.

In order to be "competitive" in garden variety math competitions (i.e., excluding very top notch ones such as Math Olympiads), the student has to have practiced on the same topic so repetitively so that s/he would not need to spend much time about such questions in the competition, quickly recognize the patterns of interest in those questions, run the algorithms in his/her mind rapidly and accurately, and move to the next question without any hangover from the previous ones.

The benefit of this approach is, obviously, being "fluent" in basics. When fluent in math, one can learn new math concepts easily, prosper in their undergrad / grad math / engineering studies, and would not be bogged down by the basic math when solving engineering problems in their workplace.

I believe there is also a downside of this approach. Practicing basics to that extent may mean focusing on a well-circumscribed, fixed (or even rigid) curriculum with less depth and breadth and little exploration, and reducing math tests into a long but quick check list.

I played Olympiads competitively as a child. My practice was very different from what you describe. It was mostly team problem-solving, or else problem-solving with a mentor, and none of the problems were ever routine or trivial. They were all past Olympiads problems, and those are tricky. 

I think different people do it differently, too. But most people solve (non-trivial) problems and read or listen to (advanced) topic discussions. 

Cheers,
Dr. Maria Droujkova

NaturalMath.com
919-388-1721

-- .- - ....

 

[Linda Fahlberg-Stojanovska]

I have been thinking about this in a slightly different context - more along the lines of "outliers", where in order to be "highly successful", i.e. competitive, in any field you must put in your 10000 hours, most of which is the going over and over the "trivial" bits until they are "first language" instead of "second language".

I think that if you are/were successful in any competitive field  (whether math, sport, blah, blah), you were "good at" and (at least initially) didn't notice that you were doing these boring trivial bits or you were directed towards it by parents, teachers, ... (I would think a lot of us at mathfuture are in the first group since we are still "enjoying" math.)

Probably most of us would agree that -at least initially- it is not wise for others to pressure a child to do his 10000 hours, which may be the question here.

To me the question is: How much pressure should we ourselves put on ourselves in order to be competitive -i.e. when does it stop being worth it? I was listening to "wait, wait, don't tell me" one week when the olympic runner jackie joyner was on and they asked her where her medals were and she answered "in a closet somewhere". when asked why -she answered that when she looked at them, she remembered all the hours and hours of training".

Does it pay to be successful in any competitive field, let alone not successful after all those hours  (there is another book by gladwell about ...most drug dealers live with their mothers...), do we want to encourage this type of dedication, can one be successful in more than one field, ...?

Warm regards to all, linda

Sent from my iPad

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[Maria Droujkova]

On Thu, Apr 2, 2015 at 1:19 AM, Linda Fahlberg-Stojanovska <lfah...@gmail.com> wrote:

I have been thinking about this in a slightly different context - more along the lines of "outliers", where in order to be "highly successful", i.e. competitive, in any field you must put in your 10000 hours, most of which is the going over and over the "trivial" bits until they are "first language" instead of "second language".

Be careful with that book, "Outliers". It's a good story, but none of the details are literally true.

I think that if you are/were successful in any competitive field  (whether math, sport, blah, blah), you were "good at" and (at least initially) didn't notice that you were doing these boring trivial bits or you were directed towards it by parents, teachers, ... (I would think a lot of us at mathfuture are in the first group since we are still "enjoying" math.)

Probably most of us would agree that -at least initially- it is not wise for others to pressure a child to do his 10000 hours, which may be the question here.

To me the question is: How much pressure should we ourselves put on ourselves in order to be competitive -i.e. when does it stop being worth it? I was listening to "wait, wait, don't tell me" one week when the olympic runner jackie joyner was on and they asked her where her medals were and she answered "in a closet somewhere". when asked why -she answered that when she looked at them, she remembered all the hours and hours of training".

I stopped playing Olympiads around 15 and switched to largely cooperative activities. I felt forced to compete for things like college entry, then chose some competitive things like grants, but tried not to put any emphasis on that in my life. For example, tenure track is out, because it is centered on all these high-stakes competitions. My overall stance in work and in business has nothing to do with competition. 

But I recognize competition as a meaningful feature of playful experiences. I am okay with using competitive mechanics for game design and experience design.

I will do competition as long as it's not Hunger Games - that is, when it's not about anything from my Pyramid of Needs.

 

[z...@unizor.com]

I totally agree with

"I will do competition as long as it's not Hunger Games - that is, when it's not about anything from my Pyramid of Needs."

When it goes to these types of competition, it's hard work that must be paid for, like in professional sports.

Back to math competition.

There is certainly a lot of good in arranging them, but we must approach them very carefully.

First and foremost, they should be directed towards the most important goal of education. What is that goal? Different teachers see it differently, and that is what differentiates one competition from another, good from bad.

If you see the purpose of math education as to drill your students in multiplication table or taking derivatives from the linear and quadratic functions, your competitions will be exactly what we are criticizing in this chain of e-mails.

If, however, you see the purpose of math education to be to develop students' creativity, logic, analytical thinking and intelligence (like I do), the competitions do have their place, they are not supposed to be similar to a military camp with a drill sergeant. They will involve interesting, non-typical and non-trivial problems to solve.

Not every student will be interested in such competitions. So be it. However, students with a spark of curiosity will take part, and these competitions will be useful for their self-esteem, their choice of future profession and will be the important component in their future success. Especially, if they win!

In addition, I have to say that I do not see anything wrong with only a minority of students being interested in taking part in competitions of this kind. Not every one is good in solving unusual problems, some are better in developing good skills doing what has been done before, just better and faster. Good for them, they will be great professionals. Different people gravitate to different kind of work and choose what's more comfortable for them. If anybody thinks that this point of view does not correspond to the general policy of "no child is left behind", I am ready to argue my point.

I am not just commenting about this, I am genuinely interested in addressing deficiency of our math education directed towards developing qualities I mentioned before, like creativity and intelligence. I was fortunate enough to receive this type of education back in the Soviet Union, and being absolutely sure in its usefulness, decided to use the Web to offer it to all. My site unizor.com is an attempt to do just that. Would appreciate any opinion about it.

Zor Shekhtman
Founder of Unizor Education 
Creative Mind through Art of Mathematics
http://www.unizor.com 

[Bradford Hansen-Smith]

Competition is a less deadly replacement for war. It supports best over the rest. Survival of the fittest. Self abuse others pay money to see.  Hunger Games is technological  extreme of fight clubs, entertainment at others expense.

Because of internet we see people doing what they do for love of doing it, the satisfaction, the esthetics, of being able to. How far one can go with something of interest; out of curiosity and love for discovery. Being gifted is not necessarily being good at something. Having enough curiosity and interest to use thousands of hours to discover possibilities of mind and body potential that has personal significance serves to move community forward. This comes from love of what is greater, not fear of being less.

When doing something to prove ourselves over others we have most probably lost before the game begins.
It must be said, Math Circles are preferable over fight club cages. The circle demonstrates value to all individual aspects, not one over the other.

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[Mehmet Kayaalp]

Competition is a less deadly replacement for war. It supports best over the rest. Survival of the fittest. Self abuse others pay money to see.  Hunger Games is technological  extreme of fight clubs, entertainment at others expense. 

 

Although I have a pretty dim view on the competition-based education and competition-centric life styles (i..e., market economy and corporate america), I am not sure I could go to the extreme and say "all competitions are bad." Most kids love to play video games. Most of those games contain some sort of competition. In other words, some competition can be pretty rewarding and/or joyous in certain contexts.

I played Olympiads competitively as a child. My practice was very different from what you describe. 

I excluded Math Olypmiads from the set of competitions that I was talking about. In Math Olympiads, as far as I know, you receive a few questions and plenty of time to think about. I am more worried about the habituation that students gain from math competitions in which students have 2 minutes or less for each multiple choice question. In that situation, the student is either familiar with the type of question or not. If not, the outlook would be pretty dire. What the student needs to do is quickly recognize the patterns that s/he studied in the past, plug-in the corresponding formula/algorithm, run the arithmetic / algebra very quickly but accurately, and fill in the right bubble.

Does it pay to be successful in any competitive field, let alone not successful after all those hours  (there is another book by gladwell about ...most drug dealers live with their mothers...), do we want to encourage this type of dedication, can one be successful in more than one field, ...?

I guess all these depend on the kid's situation, i.e., where s/he lives, the wealth of his/her family etc. Being a Renaissance man might be a joyful life style, but there is the danger that the outcome can very well be a jack-of-all-trades and master of none.

Since math is a tool set that help you excel in many fields; it has a special place in education. Therefore, I cannot easily say that it is not important not being fluent in basic math skills as long as we train the student how to think. In other words, yes I agree that math education is important in and of itself because it makes us wiser, but furthermore, these skills are also prerequisites of many other fields; thus, the more fluent you are in basic math skills, the easier you understand concepts of those fields. And a modest level of training for math competitions might help one to gain those skills much ingrained.

But what the right level is hard to judge. If you exceed that level, it may become a focus of life with diminishing returns--but only for a few lucky talents, the return could yield great success.. If you raise your competitive ambitions higher than you can achieve, frustration could emerge, which is the last thing you want.

--mehmet

[kirby urner]

On Thu, Mar 26, 2015 at 12:44 PM, Mehmet Kayaalp <mehmet....@gmail.com> wrote:

Dear All:

I am uneasy and ambivalent about the role and importance of math competitions and want to hear from you what you think about them.

Interesting thread, which I've been catching up on.

Among tournaments I've been involved in, tangentially not as a team member or judge, was a statewide programming face-off at Willamette U down valley. 

Teams from around the state wrote Project Euclid type programs on the fly, in a language of their choice, to solve provided problems. 

I was down from Portland with a coven from Free Geek (freegeek.org) to help prepare high school CS teachers for the transition from Java to Python, by now largely complete (at least in top schools).  The tournament was coupled with a CS teacher all hands, by tradition.

This Willamette U meetup provided the nucleus for a later lobby I joined, to convert CS material into for-credit math courses ala Discrete Math, a successful effort.  After high school Algebra, you now have more choices emerging to compete with the Delta Calculus track (how I refer to the traditional calc / pre-calc, in contrast the the Lambda Calculus of CS).

The tricky part at our end is to provide the right question to the student at the right time.

This was *not* done in the one "Olympiad" (named as such) I participated in as a team player. 

Each competing school was given a set of problems to work on, if not solve, and judges would ascertain the winner. 

I'm 95% sure one of those "problems" was to prove Fermat's Last Theorem though they didn't call it that.  Obviously inappropriate, but I suppose the theory was, if there was a true genius among us the chicken scratches left by that genius might be worth framing someday... I have no idea.

Where is the place of math competitions in this balance? Should competitions be part of math education? Does your answer depend on the student's type or character?

Thanks,

--mehmet

In the so-called "real world", teams compete all the time, in the military sphere sure (cite Leonardo da Vinci), but really in the commercial sphere with weapons systems first and foremost profitable commercial products with status attached. 

[ If you're Greece, you want an expensive German-made submarine purchased on credit so young Turks can strut and puff about their navy  ]

Mathematics helps teams compete on getting into space, from Sputnik to Space-X to the Chinese moon probes.

http://www.space.com/27627-china-moon-mission-earth-return.html

Schools also compete in the real world, to offer the best and/or most affordable math and/or STEM education ("math in isolation" is maybe going the way of the dinosaur except as a college subject akin to Sanskrit i.e. uber-specialized).

So we should be preparing students to life in a competitive world, sure. 

However I'm not sure the Olympics or Olympiads provide the most relevant motif. 

I know we owe a lot of our math to the ancient Greeks and kids should respect the Olympics as a good will building institution among nations (its original intent), but it's precisely the "nation state" aspect that I object to as too unreal.

Nationalism is mostly a 1900s phenomenon and is of dwindling importance.  The competing corporations are trans- or supra- national these days and just wave their various flags for marketing purposes.  Kids who grow up trying to follow the action in terms of this or that nation state versus this or that other nation state are going to be seriously confused about world affairs.  So I propose abandoning the Olympiad motif, leaving that to the Model UN (there's also Model NATO).

http://www.unausa.org/global-classrooms-model-un/how-to-participate
http://www.internationalmodelnato.org/

Kirby

[Mehmet Kayaalp]

I just found this article at http://artofproblemsolving.com/articles/competitions-pros-cons.

Enjoy!

--mehmet

Pros and Cons of Math Competitions

by Richard Rusczyk

Mathematics competitions such as MATHCOUNTS and the American Mathematics Competitions are probably the extracurricular math programs with the widest participation. The most immediate value of these math contests is obvious – they pique students’ interest in mathematics and encourage them to value intellectual pursuits. Kids love games, and many will turn just about any activity into a contest, or in other words, something to get good at. Math contests thus inspire them to become good at mathematics just like sports encourage physical fitness. Eventually, students put aside the games. By then, hopefully an interest in the underlying activity has developed.

Beyond encouraging an interest in mathematics, contests help prepare students for competition. For better or worse, much of life is competition, be it for jobs or resources or whatever. Competition of any sort trains students to deal with success and failure, and teaches them that effective performance requires practice. Moreover, nearly every interesting and worthwhile venture in life comes with some element of pressure; competition teaches students how to handle it.

Despite all the benefits of math contests, they are not an unmitigated good. First of all, not all contests are designed well. Students shouldn’t take too seriously contests that greatly emphasize speed or memorization. Curricular contests (particularly calculus contests for high school students) can also be misleading, as they deepen the misconception that there is no more to math than what is in the classroom. Such contests run the risk of encouraging students to overvalue skills that aren’t nearly as valuable as the one asset a contest should help them develop – the ability to think about and solve complex problems.

A second danger of contests is extending kids beyond their ability. Students should certainly be challenged with problems they can’t do from time to time, but if this happens consistently, the experience goes from humbling and challenging to humiliating and discouraging.

A third potential pitfall, burnout, often comes on the heels of the first two. Participants in math contests are just as much at risk of burnout as musicians or athletes. Parents, teachers, and the students themselves should be on the lookout for signs of decreased interest, and they must be willing to back off and allow the student to rediscover an interest in mathematics on his or her own. Burnout is particularly pernicious because the end result often isn’t a backlash against competition, but against math in general. Indeed, even students not involved in contests have to watch out for burnout, though the pressure of contests tends to encourage burnout more quickly than the classroom.

These possible perils are usually more than offset not only by the values we’ve already mentioned, but also by the greatest asset of math contests - cooperation. These competitions bring together students of like interests and abilities, allowing them to form their own community in which they will find friendship, inspiration, and encouragement to a far greater degree than most of these students can find in the typical classroom. Whereas a student may be one of only three or four in her school who pursues math the way others play basketball, she will not find herself so lonely at a math contest, where she’ll find many kindred spirits.

In summary, math contests are a tremendous social and intellectual opportunity for students, but exposing students to contests must be done wisely, else they become counterproductive to the goal of encouraging a lifelong interest in mathematics and other intellectual pursuits.