Hey, I wanted to pass this tidbit your way- my son’s kindergarten teacher- who does an excellent job, in our opinion- revealed confidentially to my wife that she sometimes allows some of her students to retake standardized tests if she feels they didn’t do as well as they could have.

Obviously this is something she and we are keeping quiet, because she is an excellent and devoted teacher, and if this is what it takes for her to stay afloat, we see it as more an indictment of the system than of her character.

It’s interesting to note that one of the three incorrect answers is less than 1/4 of a mile, one is equal to 1/4 of a mile, and one is greater than a whole mile.

Suppose we begin with the naive assumption that a student who takes a guess at the answer gets it correct 25% of the time.  I would hope this percentage would be higher given that an educated case, which seems more than plausible, would have a better chance of success.  Then again, a partially informed educated guess might lead to a certainty of an incorrect answer.  Still, suppose 100 students take the test, and n understand the problem and answer is correctly, while the other 100 – n guess randomly.  Our expectation is that 1/4 of the guesses are correct, meaning the number of students who bubble the correct answer is

n + (100 – n)/4 = (3/4)n + 25

When this total is 31 (as in 31% get the answer correct), that means that n = 8.

Let’s reflect on this observation.  Only 8% of Rhode Island high school juniors can recognize and compute 3 times 1/4 times 5280.  Granted, there is some context involved, but the context is simple and ordinary.  I doubt that many students can do the math who can’t comprehend the problem.

Maybe there is some variance and perhaps my naive modeling assumptions aren’t perfect, but I doubt that the number of students who truly solved the problem is particularly close to 31%.  Maybe 10% or even 15%, but not 31%.  Then again, haven’t a substantial fraction of the lowest performing students dropped out by the point this test is administered?

The article states that there is an epidemic in education, but if this problem and performance are at all indicative (it’s just one data point, but…), then math education is broken far beyond what most people imagine.  There must be entire schools where a single-digit percentage of the students can demonstrate a correct solution.

MIST Academy Award Winners on the AMC 8 by years:

2008: 8
2009: 17
2010: 16
2011: 30

I suspect this recent jump represents a jump in the number of students who have been with us since an early age, exploring a wider variety of mathematical topics to a deeper level.

I’m debating whether or not to do the same with each new MATHCOUNTS, AIME, and AMC 8 exam.  These exams have solution guides, but they usually aren’t written in a way that captures the mind of the broadest audience, and that’s my goal.

In particular, much of this curriculum should be helpful to students studying for the AMC 8 exam and MATHCOUNTS.  It does not include the harder concepts and problems tested at state or national MATHCOUNTS, but it should be accessible to a wide swath of students — both contest problem solvers and otherwise.

This curriculum is currently in what I would call “semi-polished” form.  That’s fine with us because it’s best to be practical and help students learn now than to be perfectionist and wait months or years until its in its highest quality form.

We’ll gradually release thousands of pages of curriculum at many levels over the next year or two, and continue to polish our work, adding additional features to the site along the way.

The current incarnation of the Gliya website will be short-lived.  It will evolve with some new resources over the next few months, but will change into something radically different next year.  Our goal is to leverage internet technology in targeted ways to make education easier to achieve, more enjoyable, and more accessible to students worldwide.

Our first free resource is the Gliya Network forums where I (and others) will be helping math students not only at MIST Academy, but others who join the forums as well.  Elementary, middle, and high school students are welcome to join as well as all others curious about elementary mathematics or math competitions.  We encourage parents to join as well.  See you there.

My blog-like posts will be made public.  If you want to find me on the web writing more consistently, get a Google Plus account and add me.  My gmail address is crawford.mathew@gmail.com.  I’m interested in reading your thoughts as well!

“Even failed experiments and theories teach valuable lessons.”

Not entirely disconnected, here is an email I recently received from Jason Knapp of Ohio:

For what’s it worth – I have a great amount of respect and admiration for what you’re doing, and what you have achieved with your Academy. My son loves your Number Theory book – it’s his favorite out of the AoPs set.

[anecdote for your classroom]

I have a mathematics background, and I direct research for a small biotech company. I’m honestly dismayed by the ever increasing difficulty of finding people with good problem solving skills – from finding interns to hiring PhD level scientists. I have actually recommended some of the AoPs books to some of our interns (kids working on their masters – smart, but poorly trained). Not because they need to know power-of-a-point or sum-of-cubes, but because the process of solving those types of problems is remarkably the same as that used to solve *real* problems: getting organized, searching for patterns, thinking of a simpler problem, changing your point-of-view, etc …

There certainly are some reasons why grades are beneficial, but I think that their total weight is not sufficient to justify the great costs.  I would say that the main reason that we grade is since we really are not in the business of providing an education but rather in the business of providing degrees.  To award a meaningful degree you need to grade.  To deliver a meaningful education you do not.  I know that I shouldn’t complain since the degree-awarding business (aka higher education system) is really a great one to be in, but I do wish that someone would design an education-delivery one.

To read the whole article, click here.  I’m always glad to see professors evaluating this problem.  To me the answer seems clear: divorce the process of education from the proceedings of degree/credential granting.  My hope is that at least some of my career is spent encouraging this change.  I happily spend far less than 50% (perhaps 5%) of my time grading.  We give no formal grades (though we do share some test score information during the evaluation process at MIST — when it’s useful).

*There should be problems for every interested student at the competition to work on.
*There should be problems that challenge every student to think in new ways with tools they already understand.
*Students should be interested enough in the problems themselves to want to read the solution manual (at least parts) and they should learn from reading it.
*It should be unlikely that more than a scarce few students finish more than 25 problems.
*The easier problems should take the best students very little time so that the top students can quickly move through at least half of the test.

I felt the last principle to be highly important because one hour is not a lot of time and 30 is a large number of problems.  I wanted the most motivated students to have a different competition experience where they tried to solve 6 or 8 or 10 of the 15 harder problems and there was plenty of diversity among the challenges to make places between the students highly meaningful.  I also wanted students with singular expertise to be able to earn their teams more points as opposed to writing a test with a low ceiling for perfection where many schools might earn maximum points and places are determined more by errors than solutions.  To me this is a positive definition of specialization and that seems to be what the topics tests are all about.

Here are the tests that I wrote:
* Alpha Sequences and Series (key and solution guide)
* Open Combinatorics (key and solution guide)

It’s not clear to me if they were ever edited between the time I wrote them and the time the competition was held because I never received feedback.  So it may be that the competition tests looked different.

Just teach math.  The concepts.  If you need to rely too much on gadgetry, that’s a good sign that the message isn’t being received clearly.

I suspect with good reason that the show will be very good.  While I only met Reichart once very briefly, he has a big personality and strikes me as an entertaining guy.  Garrett has become well known for his Theory of Everything — long sought after in physics — is a very comfortable public speaker.  Unfortunately I’ve been unsuccessful in finding a debut date for the show, so stay tuned.

I am very happy to report that the Alabama team’s score rose more than that of any other team in the nation.  Last year Alabama surely had the youngest team with 6 middle schoolers and only 9 high school students including just 2 seniors.  That team outscored 40% of the 120 other competing teams at the 2010 ARML contest.  This year the team was a bit older and more experienced, though still perhaps the youngest competing top team with 4 middle school students and only 3 seniors.  Alabama scored 77 points more than last year and outscored 65% of the 132 other competing teams.  An excellent showing for a still very young team.

The good: The team improved a great deal during the Power Round.  While scores on this round were higher nation-wide, perhaps no other team improved by 19 points in this 50 point round.  The relay scores were also good with Alabama finishing 10th in the nation — the highest finish during any round.  Also, the Alabama B team scored 21/50 during the Power Round which is great for a team of mostly middle school students.  Senior Owen Scott and junior Jerry Hsu found themselves in the Tie Breaker after scoring 8/10 during the Individual Round.  From what I gather this is the equivalent of scoring in the 95th to 99th percentile overall among individuals — very impressive considering the level of competition.

The bad: The team solved 5/10 during the Team Round for a total of 25/50 points.  This team was certainly capable of a better performance during the Team Round, so there is substantial room for improvement.

The fortunate: Organizing and grading the Power Round was much easier this year due to the nature of the problem.  Instead of feeling exhausted after the process I was able to enjoy watching the end of the competition — particularly the Super Relay where Alabama nearly pulled off first prize (had the answer, not simplified *sigh*) and won second prize (a large pile of candy that got passed around the bus).

Only 3 of the 30 Alabama students were seniors and it is likely that we’ll have several more of the state’s math superstars compete with us who couldn’t make it this year.

Thanks to Sergey Sarkisov and the several parents who volunteered their time to help with the team and support Alabama ARML.

The prevalence of Simpson’s paradox motivated the way I wrote the MIST Academy conditional probability lessons, which include one example of a Simpson’s paradox, then challenges the reader to unravel a storyline problem where the reader’s opinion of the data is typically anchored by the “wrong angle” and a lack of understanding of Bayesian statistics.

Natalie’s mention of her triumph spotting a Simpson’s paradox at work also helped me recognize something I’ve been doing for a while almost instinctively and without thought: I mentally correct for Simpson’s paradox or imagine potential corrections.

I recall vaguely the point at which I began to understand grammar rules well.  By “well” I specifically mean that I started spotting grammar mistakes everywhere: newspapers and magazines in particularly — not to mention advertisements and anywhere people might write in complete sentences.  As recognition of errors became more natural and automatic, it began to irritate me that I spotted quite so many errors by professional writers.  Then came acceptance, yadda yadda yadda.

The same evolution of understanding, recognition, and acceptance took place as I learned about conditional probability and analogous statistical concepts.  Now I see poor (or at least hastily accepted) interpretations of data everywhere.  It’s almost ubiquitous.  Several times while in college I spotted Simpson’s paradox in papers posted on doors in the Psychology building and offered to talk to researchers about the problem, which was the first time I really and truly got the first-hand feeling that not all scientists like to discover that they might be wrong. Though to be fair I should mention that I’ve had much better conversations with most scientists about these kinds of data problems.

Here’s an example from a professional economist:

Does that mean that college isn’t worth it? Not exactly. In fact, given the crappy economy, a college degree is more valuable than ever, a point that Levitt makes in a recent Freakonomics Radio podcast.  The most telling statistic as to the value of college: the unemployment rate among college graduates is less than half (4.5%) than people with only a high school diploma (9.7%). (See the BLS employment status table here.)

There may be arguments in favor of the high tuition costs at universities, but this one fails hard, and it’s disturbing to see a highly noted economist (and one whose works I enjoy) use it as the “most telling statistic”.  Other variables need to be considered.  On average, college graduates have higher IQs, more money, and better understanding of the structure of the society they live in to begin with than those with less education, so there is a clear selection bias.  If college involved involved nothing but football games and keg stands we would expect college graduates to achieve higher levels of employment.  Even the nature of the jobs considered matters.  Recently the increase in minimum wage knocked over a million jobs out of the economy.  Few if any of those jobs included college educated workers, but plenty included those with a high school diploma, setting up a clear conditional that must be resolved before the statistics above are particularly meaningful.  Really, these only begin to scratch the surface of my best arguments that important conditionals are missing in this case — they’re just the ones that are easiest to point out in one paragraph.  A whole book could be written on the conditionals that affect the above statistics and its damning to the economics community that few economists seem to recognize that. This point should at the very least be mentioned, and the moment it gets mentioned the economist would have to step back and reevaluate the story told by the statistics.

It may certainly be the case that college is economically worth the high costs, but any assertion that these statistics anchor the debate is absurd.

I see it everywhere.  Eventually I fell prey to acceptance.  I see it everywhere.  I sometimes complain about it to anyone who will listen (not that often).  I do nothing about it.

Well, I did eventually write some really cool classes about it for my students.  Now it’s a standard part of the curriculum I use with students as young as 10 years old.  These really are some of the more creative classes I’ve written, so I’m particularly proud of them.  But the reason I spent so much time thinking about them (and plan to spend even more in the near future) is because I’m shaking off the acceptance that the world needs to be this way.  It doesn’t.  And there is opportunity!

People will always misinterpret statistics, but motivated problem solvers can find ways to minimize the impact of the problem.  I once considered building a medical data consulting service that I figured would be based primarily on hunting down and extinguishing the damage caused by Simpson’s paradox and analogous Bayesian misunderstandings in health care/medicine — which is quite extensive and costly both in terms of dollars and lives!  I’ve decided to both blog about this idea and encourage others to take up this cause professionally so that I don’t have to change vocations to make a difference.

Health care is a slam dunk example.  You’d have a hard time convincing me that we can’t save billions of dollars annually in health care costs if we set some really smart people about the task of ferreting out all the problematic interpretations of data.  But there are examples of damaging data misinterpretations all over the world — including particularly valuable problems to solve in the business world.  In fact, an old friend of mine just found one…