Math thread

Maybe we should move the math talk over here, rather than continuing the Cornell-Clarkson thread ad infinitum.

Also, it gives me a place to post this.

And this:

Math is just one theory, and I think schools should be allowed to teach alternative theories as well.

jtwcornell91
Maybe we should move the math talk over here, rather than continuing the Cornell-Clarkson thread ad infinitum.

Also, it gives me a place to post this.

Slide 6.

Nice post. Us liberal arts types hustled over to Wikipedia to look up Riemann-Zeta Function. More than the math geeks had to go look up herpes.

Friend of mine just posted this on Facebook. Liked it. Linked it.
Math is not linear.

Robb
Friend of mine just posted this on Facebook. Liked it. Linked it.
Math is not linear.

This is great, thank you for posting it.

Robb
Friend of mine just posted this on Facebook. Liked it. Linked it.
Math is not linear.

I like this very much, too. One major obstacle to the approach advocated here is that it requires math teachers who are comfortable with a broad base of math concepts, not just the linear material. I fear that many are well-intentioned, but just not sufficiently well educated. I remember parents night, or whatever they called it, when my daughter's math teacher apologized for having to spend a week on polar coordinates, which he said were totally useless, just because they're in the book. I refrained from screaming at him, but afterward told him that I disagreed with that assessment. He acknowledged that he had heard that they were used in electrical engineering, but as his college major was economics, he had not had any exposure to their applications. A couple of days later I sent him a couple of examples of ways in which I had used them recently. Two years later, when my second daughter had the same class, he was using my notes as handouts.

David Harding

Robb
Friend of mine just posted this on Facebook. Liked it. Linked it.
Math is not linear.

I like this very much, too. One major obstacle to the approach advocated here is that it requires math teachers who are comfortable with a broad base of math concepts, not just the linear material. I fear that many are well-intentioned, but just not sufficiently well educated. I remember parents night, or whatever they called it, when my daughter's math teacher apologized for having to spend a week on polar coordinates, which he said were totally useless, just because they're in the book. I refrained from screaming at him, but afterward told him that I disagreed with that assessment. He acknowledged that he had heard that they were used in electrical engineering, but as his college major was economics, he had not had any exposure to their applications. A couple of days later I sent him a couple of examples of ways in which I had used them recently. Two years later, when my second daughter had the same class, he was using my notes as handouts.

Ha. Well, this presentation would be how to teach math excellently. Sadly, very few people in any field are really destined for excellence...

My math teacher was great - more than 10 out of our class of 20 got 5s on BC calculus. However, even she was largely unaware of the practical applications for many of the topics. In my experience, high school teachers tend to be teachers first, and subject matter experts second. Clearly, a balance is needed.

Since it's anecdote time, yours reminded me of my physics teacher, who was terrible. Almost nobody ever even bothered to take the AP exam. I worked my butt off for 6 weeks (spending upwards of 20 hours per week on my own) to get myself ready for the exam. Three years later, when my sister took the class he was introducing himself and said something along the lines of, "I think I do a pretty good job. I even had one student who got a 5 on the AP exam..."

My high school physics teacher was also terrible. He quit teaching a couple years after I graduated to become a priest

This conversation reminds me a bit of a suggestion I heard on NPR a while back on reforming universities. The person pointed out that professors in universities are often extremely specialized; they'll know their one (sometimes esoteric) field through and through, but most useful applications require application of various fields. The person's suggestion was that majors be focused on problems rather than specific academic fields; one might major in water shortage problems in poor countries, for instance -- rather than in biology, or politics, or economics, or operational engineering, or any of the other fields that go into solving water shortage.

I don't think the idea should be taken to its full extreme, but I thought it was an interesting idea nonetheless.

ftyuv
The person's suggestion was that majors be focused on problems rather than specific academic fields; one might major in water shortage problems in poor countries, for instance -- rather than in biology, or politics, or economics, or operational engineering, or any of the other fields that go into solving water shortage.

I don't think the idea should be taken to its full extreme, but I thought it was an interesting idea nonetheless.

Interesting idea, though one problem comes to mind. Generally, we organize work around a problem and then call in experts from a lot of fields who bring field-general principles to bear on a reality-specific phenomenon. Those experts are useful for an infinite number of "accidental" problems in real life.

The suggestion strikes me as training architects to build the Sears Tower. You'll wind up with a great Sears Tower when you're done, but not much else.

And for that matter, if you already know enough about a problem to teach to it... it's not really a problem.

But it's probably a decent perspective to keep in mind.

Trotsky

ftyuv
The person's suggestion was that majors be focused on problems rather than specific academic fields; one might major in water shortage problems in poor countries, for instance -- rather than in biology, or politics, or economics, or operational engineering, or any of the other fields that go into solving water shortage.

I don't think the idea should be taken to its full extreme, but I thought it was an interesting idea nonetheless.

Interesting idea, though one problem comes to mind. Generally, we organize work around a problem and then call in experts from a lot of fields who bring field-general principles to bear on a reality-specific phenomenon. Those experts are useful for an infinite number of "accidental" problems in real life.

The suggestion strikes me as training architects to build the Sears Tower. You'll wind up with a great Sears Tower when you're done, but not much else.

And for that matter, if you already know enough about a problem to teach to it... it's not really a problem.

But it's probably a decent perspective to keep in mind.

I think this actually already occurs in a number of fields. For example, my sister-in-law is working on a masters of public health degree. Her courseload is incredibly diverse - everything from pure statistics to virology to economics to vaccine policy to international relations. It's a real mashup of all the tools and knowledge that it would take to be a proficient problem solver in the field of public health. This certainly doesn't bring to mind your overly-specific "Sears Tower" fear. In fact, it's really the exact opposite - they're delving into such a wide variety of topics that they aren't being trained how to solve any specific problem.

Robb

Trotsky

ftyuv
The person's suggestion was that majors be focused on problems rather than specific academic fields; one might major in water shortage problems in poor countries, for instance -- rather than in biology, or politics, or economics, or operational engineering, or any of the other fields that go into solving water shortage.

I don't think the idea should be taken to its full extreme, but I thought it was an interesting idea nonetheless.

Interesting idea, though one problem comes to mind. Generally, we organize work around a problem and then call in experts from a lot of fields who bring field-general principles to bear on a reality-specific phenomenon. Those experts are useful for an infinite number of "accidental" problems in real life.

The suggestion strikes me as training architects to build the Sears Tower. You'll wind up with a great Sears Tower when you're done, but not much else.

And for that matter, if you already know enough about a problem to teach to it... it's not really a problem.

But it's probably a decent perspective to keep in mind.

I think this actually already occurs in a number of fields. For example, my sister-in-law is working on a masters of public health degree. Her courseload is incredibly diverse - everything from pure statistics to virology to economics to vaccine policy to international relations. It's a real mashup of all the tools and knowledge that it would take to be a proficient problem solver in the field of public health. This certainly doesn't bring to mind your overly-specific "Sears Tower" fear. In fact, it's really the exact opposite - they're delving into such a wide variety of topics that they aren't being trained how to solve any specific problem.

I'm not going to argue against people getting training outside their cubbyhole; that's always good.

Trotsky

ftyuv
The person's suggestion was that majors be focused on problems rather than specific academic fields; one might major in water shortage problems in poor countries, for instance -- rather than in biology, or politics, or economics, or operational engineering, or any of the other fields that go into solving water shortage.

I don't think the idea should be taken to its full extreme, but I thought it was an interesting idea nonetheless.

Interesting idea, though one problem comes to mind. Generally, we organize work around a problem and then call in experts from a lot of fields who bring field-general principles to bear on a reality-specific phenomenon. Those experts are useful for an infinite number of "accidental" problems in real life.

The suggestion strikes me as training architects to build the Sears Tower. You'll wind up with a great Sears Tower when you're done, but not much else.

And for that matter, if you already know enough about a problem to teach to it... it's not really a problem.

But it's probably a decent perspective to keep in mind.

It's hard to argue that we don't need both the specialists and generalists. The trick is finding the right balance, even within a field like medicine. One of the beauties of Cornell is the breadth of really deep expertise that can be brought together.

Jeff Hopkins '82
My high school physics teacher was also terrible. He quit teaching a couple years after I graduated to become a priest

Now the boys in the parish will be nervous about being around physics, too.

Any chance we could get a Krach graph of college football for this year?

French Rage
Any chance we could get a Krach graph of college football for this year?

Not in graphic form but:
[mattcarberry.com]
[www.vaporia.com]
It's interesting how the two versions differ as they use different weightings for the dummy game necessary to avoid having teams with an infinite rating.

jkahn

French Rage
Any chance we could get a Krach graph of college football for this year?

Not in graphic form but:
[mattcarberry.com]
[www.vaporia.com]
It's interesting how the two versions differ as they use different weightings for the dummy game necessary to avoid having teams with an infinite rating.

It's also a little insane how much better Auburn's rating is than everyone else (in both rankings). Matt makes Auburn a 6:1 favorite over Stanford, while Vaporia has them at a whopping 49-1. I'd take either of those bets straight up!

Robb

jkahn

French Rage
Any chance we could get a Krach graph of college football for this year?

Not in graphic form but:
[mattcarberry.com]
[www.vaporia.com]
It's interesting how the two versions differ as they use different weightings for the dummy game necessary to avoid having teams with an infinite rating.

It's also a little insane how much better Auburn's rating is than everyone else (in both rankings). Matt makes Auburn a 6:1 favorite over Stanford, while Vaporia has them at a whopping 49-1. I'd take either of those bets straight up!

Actually, Matt has Oregon ahead of Auburn, and they'd be 49-1 vs. Stanford. Vaporia (John Wobus) has Auburn at #1 and 6:1 vs. Stanford. While I think these are the the best way to order ranking based on results, the small sample size plays havoc with the ratios.

Interesting that Carberry has the Big 12 as the best conference despite not having a member in the top 8. I guess it's like golf -- have the best worst hole.

Trotsky
Interesting that Carberry has the Big 12 as the best conference despite not having a member in the top 8. I guess it's like golf -- have the best worst hole.

jkahn
the small sample size plays havoc with the ratios

Greg is right here. The conference rating is simply the average of the non-conference RRWPs of the conference's teams (listed under 'nCWP' in my tables). The Big 12 has eight members in the top 25 and all twelve are in the upper half of I-A. Three SEC teams are rated worse than the lowest Big 12 team, Kansas. The Jayhawks' SOS is also the lowest in the league, at 38.

This highlights another problem with the application of KRACH to college football - the "Appalachian State problem." Kansas, in particular, would likely be ranked singificantly lower if its season-opening loss to North Dakota State were accounted for. Virginia Tech is now at #20, but they were in the top ten earlier in the season, because the James Madison loss didn't impact them. The Colley Matrix (a BC$ component) uses a novel method to account for FCS teams - grouping the FCS teams until they "look" like FBS teams.

jkahn
Actually, Matt has Oregon ahead of Auburn, and they'd be 49-1 vs. Stanford.

The top two flip-flopped after Saturday's games, and it had nothing to do with what either of them did - it was caused by LSU's loss to Arkansas. That dropped LSU into the "main field" of teams.

Same ratings as linked above, sorted by conference.

The biggest thing to note is that a team currently ranked #71 (#66 in the Wobus KRACH) is one win away from an $18 million payout.

I've wondered for the past few years what the NHL standings would look like using KRACH but I don't know how to set that up myself. Anyone care to point me towards a link that could explain the process to me?

kingpin248

Trotsky
Interesting that Carberry has the Big 12 as the best conference despite not having a member in the top 8. I guess it's like golf -- have the best worst hole.

jkahn
the small sample size plays havoc with the ratios

Greg is right here. The conference rating is simply the average of the non-conference RRWPs of the conference's teams (listed under 'nCWP' in my tables).

I don't see any relationship between my comment and Greg's comment here. My comment on the ratios and small sample sizes was that given the small number of games on the college football schedule, a ratio such as 49:1 Oregon vs. Stanford certainly doesn't meet our realistic expectation of the true odds, even though it's an appropriate way to establish a ranking order of the teams. It's also strongly influenced by having on of those teams being undefeated. With a 30 game college hockey schedule, we see the ratios getting much closer to our sense of reality.

jkahn
I don't see any relationship between my comment and Greg's comment here.

True. My bad.

nr53
I've wondered for the past few years what the NHL standings would look like using KRACH

I do the pro leagues too! 2010-11 NHL.
The records given in this table don't match the NHL standings; they are W-L-T, and do not include shootout results or account for the overtime loser point.

Being a part-time math geek when I'm not being a hockey geek, I was asked by email today the following question:
What are the odds of the Packers and Bears 1) both being in the playoffs and 2) playing each other in the playoffs and 3) playing each other in the NFC championship game, assuming each NFL team has a 50% chance of winning each game?

While I quickly responded with the following to the fellow who asked, I thought the question was thought provoking enough that readers of this thread might enjoy a shot at coming up with a more complete answer.

My quick reply:
"First, the easy part.
Let's figure the odds of both the Packers and Bears being in the playoffs.
Since there a 4 teams in the NFC North, and the division champ automatically makes the playoffs, there's a 50% that one of the Bears and Packers will be division champs.
Then there are two spots left for wildcards and 12 non-division champs. So if the Bears or Pack are champs, the non-champ has a 2/12 chance of making the playoffs. So the odds of both making the playoffs, one as champ and the other as wildcard, is 1/2 times 2/12 = 1/12.
Now actually it is slightly higher than that, because even if the Lions or Vikes finish first, it is remotely possible that the Bears and Pack could be the two wildcards, but that's unlikely and requires more math than I have time to spend - since it's not just a question of each team having an independent chance - if one team from a division gets the first wildcard, it's less likely that another team in that division will get the second, because that team needed an above average record, which means its wins may be losses for other teams in the division.
The same issue with dependent variables clouds the next part of the analysis. If the Bears at division champ had an equal chance to be seeded anywhere between 1 and 4, and the Pack as wildcard had an equal chance to be either 5 or 6, we could do the analysis. For instance, then 25% of the time the Bears would be #3 and 50% of the time the Pack would be #6, giving them a 1/4 times 1/2 chance or 1/8 of being seeded 3-6 and meeting in the first round. You would then multiply that by the 1/12, and there's be a 1/96 chance that one would be seeded #3 and one #6 and meet in the first round. Similarly, a 1/96 chance of being #4 and #5, so a total of 2/96, or 1/48 of meeting in the first round. However, that analysis is faulty, because it assumes that the records of the Pack and Bears are independent variables. However, we logically know that it is much less likely that two teams from the same division will be the 4 and 5 seeds than the odds of them being #3 and 6. That's because the 4th seed is the champ with the weakest record and it is likely the top wildcard will have a better record and not be from that division. So the analysis gets very hairy at this point.
Perhaps more later if I have time."

jkahn
What are the odds of the Packers and Bears 1) both being in the playoffs and 2) playing each other in the playoffs and 3) playing each other in the NFC championship game, assuming each NFL team has a 50% chance of winning each game?

The odds as of today are 1.

Lets ask a more challenging question: what are the odds of Bengals and the Browns meeting in the AFC Championship game? Somehow I think there's a better chance that people can sense the future through psychic powers...

(Implicitly rejecting the coin flip assumption.)

Super Bowl Squares odds. (warning: url is for a gambling site that might be flagged by some work browsers so follow at own risk)

Here is a table with the results from almost 20 years of NFL games. I assume since this is a neutral site game one is better off taking the average of each {x,y} and {y,x} pair. The precedence of numbers (7, 0, 4, 3...) is not that surprising.

Nerd crack.