David Croushore

A Man in Progress

Past Perspective

I wonder how common this phenomenon is, but I occasionally hear ideas that I’ve believed in the past and find that they sound very strange.  As an example, when I was about 22 years old, I was convinced of the fact that I would never get married, and I had all kinds of justifications for that position.  Hearing those same justifications today, they sound very hollow, and I have a difficult time understanding the perspective of someone saying them.  But I used to think that way!  How can something that was generated from the depths of my own brain sound so foreign that I cannot even imagine the circumstances that would lead to that perspective? 

I can’t decide whether to embrace this lack of perspective, or fear it.  On the one hand, it seems that being unable to recreate the mental conditions that I used to occupy shows evidence that I am actually capable of changing my mind (something that people do far less than they think).  On the other hand, perspective-taking is an important skill for all sorts of social interactions, and if a perspective I once held seems inaccessible, then perspectives that are equally distant from my own, but also completely foreign, must be even harder to access.  How can I understand the viewpoint of someone who holds a view that I’ve never experienced if I can’t even understand some of the views I used to have? 

Admittedly, sometimes not being able to take someone’s perspective is fine - especially when the views they espouse are downright crazy, or even dangerous.  Unfortunately, this line of thinking borders on becoming a Fully Generalized Counterargument - “I don’t understand this person’s perspective, and believe they are wrong, therefore they are wrong.” - that doesn’t smell like a healthy, rational line of thought.  More compelling would be the ability to see the evidence from their perspective and either find the mistake or change my own mind.  The inability to access perspective prevents that.

I want to ask, “Am I the only one who has experienced this?” but the phrase “Am I the only one” is usually met with a flat “no.”  So how many people have experienced this, and what are your thoughts on it?  What did you once believe that you now find so foreign that you cannot understand how you used to think it?  And how do you feel about that state of affairs?

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Updating Probabilities Based on Repeated Observations

Conditional probabilities can be tricky, especially when you have repeated observations rather than just isolated ones.  This problem is worth working through by hand to develop an understanding for how these things work, as it demonstrates how tricky conditional probabilities get when you introduce repetition.

The context is simple: there are 2 bowls of cookies, each with 12 cookies in them.  The breakdown of cookies is as follows:

Bowl 1: 75% vanilla cookies, 25% chocolate cookies (9 vanilla, 3 chocolate)
Bowl 2: 50% vanilla cookies, 50% chocolate cookies (6 of each)

Problem #1 (easy): If you grab a cookie randomly from one of the bowls, and the cookie is vanilla, what is the probability that it came from Bowl #1?

Problem #2 (harder): If you grab three cookies from the same randomly chosen bowl, and they are (in order) vanilla, chocolate, and vanilla, what is the probability that they came from Bowl #1 if you replaced the cookie after each drawing (sampled with replacement)?

Problem #3 (hardest): If you grab three cookies from the same randomly chosen bowl, and they are (in order) vanilla, chocolate, and vanilla, what is the probability that they came from Bowl #1 if you ate each cookie after drawing it (sampled without replacement)?

Solutions below.

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Problem #1: This is a straightforward computation.  All of the needed inputs are given, so it’s as easy to solve as applying Bayes Theorem:

P(Bowl 1 | vanilla) = P(vanilla | Bowl 1)*P(Bowl 1)/P(vanilla) = (.75*.5)/.625 = .6      (Note: The .625 is calculated as P(vanilla | Bowl 1)*P(Bowl 1) + P(vanilla | Bowl 2)*P(Bowl 2) as there is no joint probability, thus P(vanilla) = .75*.5 + .5*.5 = .375 + .25 = .625)

Problem #2: This version gets trickier, because after each draw, the prior probability of drawing from either bowl has changed.  This also affects the prior probability of choosing vanilla or chocolate, though this is not obvious at first.

After step 1, which we solved above, we have P(Bowl 1) = .6, therefore P(Bowl 2) = .4  [from this point on B1 = Bowl 1, B2 = Bowl 2, V = vanilla, and C = chocolate]

Thus, the posterior probability after the second selection is P(B1|C) = (P(C|B1)*P(B1))/P(C), but P(C) is no longer .375, because P(B1) no longer equals P(B2) = .5.

P(C) = P(C|B1)*P(B1) + P(C|B2)*P(B2) = .25*.6 + .5*.4 = .15+.2 = .35

Therefore: P(B1|C) = (.25*.6)/.35 = .4286

This probability then becomes the prior probability for the third trial, and we need to recompute P(V) based on the new prior (as the decimal has gotten messy, you can choose to take my word for the following, or calculate it yourself):

P(V) = P(V|B1)*P(B1) + P(V|B2)*P(B2) = .6071

and P(B1|V) = (P(V|B1)*P(B1))/P(V) = .5294

(An interesting observation is how much more information there appears to be in the second selection than in the first or third.  To examine the magnitude of evidence, you can compare the logs of prior and posterior odds ratios.  In this case, the initial condition gave no evidence in either direction, or odds of 1:1, so ln(1:1) = 0, showing the neutrality of the information.  After the first vanilla cookies, we now had an odds ratio of 3:2, or 1.5, with a natural log of .405, an increase of .405 from the initial condition.  After the chocolate cookie, we have log-odds of -.29, a loss of .69!!  The next vanilla cookie only brings the log-odds back to .12, gaining another .405.  In other words, every vanilla cookie is worth .405 log-odds, while every chocolate is worth -.69, so the chocolate cookie has more information in it than the vanilla from the perspective of bowl 1.)

Problem #3: This version is the hardest because the probabilities change even more.  The sampling without replacement introduces additional complexity, though the mechanics of solving the problem are identical.

We’ve done step one, so P(B1) = .6 going into step two.  But not, there are only 11 cookies in “either” bowl (because we don’t know which one we sampled the first cookie from).  The updated probabilities are P(V) = P(V|B1)*P(B1)+P(V|B2)*P(B2) = 8/11*.6 + 5/11*.4 = .727*.6 + .455*.4 = .618, so P(C) = 1-P(V) = .382.

We have, P(B1|C) = (P(C|B1)*P(B1))/P(C) = .4286.  This is actually the same probability obtained the second problem where we sampled without replacement, which is an interesting result.  I’m not 100% sure of why that worked out the way it did, but it makes sense that the evidence so far (V then C) would not be affected much by the replacement since it dilutes the V:C ratio of bowl 2 as well as bowl 1. 

As we move to step 3, we now have 8/10 vanilla cookies in Bowl 1, while Bowl 2 is back to 50/50.  As a result, the new probabilities are:

P(V|B1) = .8
P(V) = .8*.4286 + .5 *.5714 = .6826
So P(B1|V) = .5455

We see that when sampling without replacement, the third cookie gives more evidence that it did when sampling with replacement.  To be sure, let’s check to the log-odds.  Since the first two steps gave the same results, the log-odds are still the same as well.  +.405 then -.693.  The third step, however, gives more evidence that it did previously, because the conditional probabilities P(V|Bx) have changed.  The final vanilla cookies now gives +.47 log-odds of evidence instead of just .405 for the Bowl 1 hypothesis. 
 

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Fun With Probability - Guessing Children’s Genders

I’m a sucker for unexpected results in probability theory.  For example, I’ve recounted the Monty Hall problem to more audiences than I care to count (yet the prior probability that any of them gets it right the first time remains near zero).  But old material gets boring, so today I’m going to share another fun probability quirk that most people get wrong.

First, the easy version:

A mother has two children, and you ask her if either of them is a boy.  She answers yes.  What is the probability that she has two boys?

Next, the harder version:

A mother has two children, and you ask her if either of them is a boy born on a Tuesday.  She answers yes.  What is the probability that she has two boys?

Think about your answers to both of these questions before you read on (assume that boys and girls are equally likely, so p(boy) = p(girl) = .5).

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It may come as a surprise to you that not only is the answer to both of these questions not p(2 boys) = .5, but that the two questions have different answers, and that those answers have absolutely nothing to do with any metaphysical properties of Tuesday.  

By way of explanation, had you asked the mother “Is your oldest child a boy?” the affirmative response would have indeed indicated p(2 boys) = .5.  That’s because you have really narrowed the question down to simply guessing the gender of the younger child.  However, because the questions merely asked about the existence of a single boy, regardless of order, the evidence received is not as strong.  Given the prior knowledge of two children, the prior probability of having 2 boys was 1/4 (there are four combinations, BB, BG, GB, and GG).  Knowing that at least one of them is a boy eliminates only one of these possibilities (GG), thus, the posterior probability, p(2 boys | at least 1 boy) = 1/3.  This can also be solved using Bayes’ Theorem, which says p(A|B) = p(B|A)*p(A)/p(B) =>  p(2 boys | at least one boy) = p(2 boys)*p(at least one boy | 2 boys)/p(at least one boy) = 1/4*1/(3/4) = 1/3.  (It is my sincere hope that it is obvious why p(at least one boy | 2 boys) = 1).

The second problem is similar, except in this case we have added another dimension (the weekday on which one of the children was born), which provides slightly more evidence.  It is natural to wonder why the Tuesday has any bearing on the result, but it turns out that it does.

Assuming that both genders and all days of the week are equally likely (both assumptions that are demonstrably false, but bear with me), consider the full set of possibilities that could exist prior to learning of the existence of B_T (boy on Tuesday).  Each of the four possibilities from the first problem has expanded to 49 possibilities (B_M, B_M; B_M, B_T; etc.).  Thus, we have a total of 196 possibilities, so the prior probability of two boys is… 1/4 (49/196), the same as it was in the first problem.  If that seems anti-climactic and obvious, you are right.  Where it gets interesting is when we consider how many of the 196 possibilities were eliminated by learning B_T.  In fact, almost all of them are.  The remaining probabilities include 7x(B_T, G_x), 7x(G_x, B_T), 6x(B_T, B_!T), 6x(B_!T, B_T), and 1x(B_T, B_T).  It is the last of these elements (B_T, B_T) that often gets double counted.  As the other examples show, the reversal of order implies a new possibility, but in the case of at least one B_T, the reversal of order does not, as there is no new evidence of (B_T, B_T) gained by learning that B_T is first or second.  As a result, the probability of BB in this example is 13/27, which is higher than the 1/3 obtained without the day of the week.   Learning the day of the week added 4/27ths to the probability, which is not inconsequential.  

Note, this version can also be solved using Bayes’ Theorem; p(2 boys | B_T) = p(2 boys)* p(B_T | 2 boys)/p(B_T) = (1/4)*(13/49)/(27/196) = 13/27, but the recognition that p(B_T | 2 boys) = 13/49 rather than 14/49 is still the key to getting it right.

A good lesson to remember the next time you hear some seemingly irrelevant piece of information.

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A Kickstarter Campaign You Shouldn’t Miss

As anyone who reads this blog knows, I’m into high brow humor.  I’m so high brow, I thought about applying for a job at The New Yorker (and I’d link to the Family Guy clip about no one at The New Yorker having an anus if I could find it, but I can’t, so thank the DMCA).

But I want to interrupt your normally scheduled programming to let you know about a kickstarter campaign that you should back in the name of less-than-high-brow humor.  The Weekend Pilots, a comedy rock trio out of LA, are offering you the chance to have a comedy song written about you, and all you have to do is contribute $25 to their kickstarter campaign for a new music video that will be 1 part Eric Prydz and two parts cocaine (literally, look at the campaign site).

For those of you who aren’t familiar with The Weekend Pilots, here’s a sneak peak at what their music videos have looked like in the past (NSFW - fair warning):

High brow indeed!

(Disclosure: One of The Weekend Pilots, Jay, was the best man at my wedding - take from that what you will)

Here’s the link to the campaign one more time:

http://www.kickstarter.com/projects/1850905868/you-need-more-music-video

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On Milestones

On Saturday, I finished my MBA.  For the past 33 months, I’ve been calling myself a part-time MBA student.  I’ve been enrolled in classes, following an academic calendar, and thinking about how to balance work, family, school, and hobbies.  Most importantly, I’ve been looking forward to a milestone, to the day when I wouldn’t call myself an MBA student anymore - to a day that turned out to be Saturday. 

I’ve always had an abstract notion of what it would be like to be finished with the MBA.  I joked with classmates about the amount of free time I’d have to fill.  All of those thoughts were wrong, and free time really isn’t something I believe in.  Being done means something really significant, but it isn’t about the credential, and it isn’t about the change in lifestyle.  Being done means not having the goal to rely on anymore. 

Progress is always easier when there is a goal at the end.  Adding structure to the pursuit makes it easier still.  The MBA fit both of these criteria.  The goal was clear, and the structure was imposed by the school and by the professors.  Making progress was easy.  Having finished the MBA, I need a new goal, something to continue to motivate the same kind of progress.  And I need to build up a new structure for the pursuit.  Those things won’t be easy.

I work best when I can harness momentum.  Getting started on something new is the most difficult task for me.  Once things start, each day builds momentum for the next.  I’ve just finished a long, and I hope worthwhile pursuit.  I need to harness that momentum and direct it towards something new.   

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The Most Asinine Rule in Sports History

Picture this - you’re watching a football game between the Denver Broncos and New England Patriots.  It’s January 2014 in Denver, there’s a strong wind but no snow.  Both defenses have played well, keeping the score modest, with Peyton Manning and Tom Brady reaching just 18 and 20 points respectively.  With 1:15 left in the game, a trip to the Super Bowl on the line, Peyton Manning’s Broncos take over at their own 30 yard line, with no timeouts, and drive down the field.  On fourth down from the 45 yard line, they attempt a 62 yard field goal, a kick that hits the crossbar and bounces straight up in the air before hitting it a second time, pausing for a moment as if to rest perfectly balanced before falling, by the slimmest of margins, through the uprights.  Somewhere in Hollywood, a director has a heart attack realizing he has missed this opportunity for the ultimate sports cliffhanger.  There are 10 seconds left on the clock and the Broncos now lead by 1. 

The Patriots have a time out, use it, and get the ball at the Broncos 25, hitting an easy game-winning field goal.  Everyone in the audience is disappointed at the lackluster finish.

Sound odd?  Luckily, that rule doesn’t exist in football.  But it does exist in basketball.

In basketball, calling a timeout after your opponent scores is an obvious end-game strategy.  It forces your opponent to concede 75% of the floor with no time ticking off the clock, and no chance to play defense.  As far as I can tell, the only purpose of this rule is to make the game less fair, less interesting, and less amenable to defense as a strategy.  What type of illogical organization would come up with, much less mandate, a rule that makes the game less fair and less interesting (conceding for the moment that defense isn’t a strategy anyone is considering in today’s NBA regardless of this rule)? 

I want basketball to have a renaissance.  I want to love that game like I did in the 90s, before the league became a collection of 4 or 5 star-studded rosters and a bunch of garbage.  Back when superstars stayed with one team and rivalries could bubble up in series after series.  Jordan lost the the Pistons before he beat them.  Magic and Bird had an epic rivalry.  Since the Celtics created the “Big Three” the league will never be the same.  Miami is the current collection of stars too afraid to play against each other.  The Clippers look like they want to be next.  It makes the sport so boring.

The NBA could learn a lot from the NFL.  Salary caps and revenue sharing do make sports more interesting.  Maybe even one up them and give your players a real pension to compensate them for the lower salaries they have to take in the short term. 

The NBA used to be the king of American sports.  It had the highest revenues by far.  Now, the NBA is third in revenue (football and baseball are ahead), and fifth in attendence (behind the other majors and MLS - yes, MLS).

Maybe having terrible rules that ruin the endgame is part of the problem. 

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The Right and Wrong Questions

“What should I do?” is usually the wrong question to ask. It imposes the burden of the decision on someone else.  In the context of a team, it’s always the wrong question.  It delegates the responsibility for your actions to someone else.  It’s the antithesis of working by subtraction.  In the context of asking for advice, it puts the other person in the awkward position of having to make a recommendation, usually without all of the background knowledge that they need.  If you do what they say, and it doesn’t work, they know you’ll hold it against them.  That’s a terrible place to put someone.

In a team, ask “How can I help?” It gets at the same issue, but it lets the other person choose whether they want your help, and let’s them find the margins of the subtractive process and give them to you.  It says “I already have a handle on my own responsibilities, but I’ve got some extra time/resources, and I want to help the team?”  It doesn’t say “I want to do the minimum because I am self-centered, so please define the boundary of the minimum for me” the way the first question does.  

When asking for advice, ask “What do you think?” It frees to advice-giver of the responsibility of issuing a recommendation for action.  It gives them an out (“Wow, that’s a tough situation, I have no idea what you should do”).  

Small changes - but the implications are big.  Asking the wrong question, even when it is well-intentioned, can be a big mistake.  Leave “what should I do?” off the list of candidates.

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Reacting to Jason Collins

It’s about time. 

It’s about time that the sports world caught up with everyone else, and someone had the gumption to admit that homosexuality exists in every world, sports or otherwise.  Sports is this weird world where the rules that the rest of us follow don’t exist.  Corruption in college sports is encouraged.  Cheating is shrugged off.  The rules just don’t apply.  Try telling a talented scientist that they aren’t allowed to take a job until this spend a year as an unpaid servant.  That’s what we do to stand-out high school basketball players.  Offer an MBA student cash to stay in school an extra year instead of signing on with a big company and getting paid to do what they do.  It would never work, but in college football it does. 

So no one should be surprised that in a world that follows its own rules, and not the rest of society’s, no gay man has had the courage to come out until now.  It has somehow remained socially acceptable to be a homophobe as long as you do it in the context of professional athletics.  So it’s about time the sports world caught up with the rest of us in at least one dimension.

But I’m afraid. 

I’m afraid that Jason Collins missed his opportunity to make a real difference.  He’s a 34 year old mediocre free agent.  It’s entirely possible, whether he came out or not, that his playing career is over.  There will be no way to tell, if no one picks him up in the off-season, whether there was any discrimination, or if he’s just not worth a contract.  We’ll never know, and that’s a problem.  Somewhere out there, there’s a twenty-something talented player who hasn’t come out.  He’s afraid of the ramifications.  If he came out, and no one picked him up, we’d all know exactly what that meant.  It would tear the veil right away from our eyes and we’d know, with certainty, just how bad homophobia is in sports. 

Of course, there’s a flip side.  There just might be a GM somewhere in the NBA who wants to make a point, who might pick up Collins just to show that discrimination isn’t a problem in basketball.  It seems appropriate that the movie 42 is out in theaters right now.  One GM making a stand against discrimination changed baseball - it could do the same for basketball.  But Jackie Robinson was a superstar.  Jason Collins is a journeyman averaging less than 2 points per game for the last 5 years, who has played for 6 teams (3 in the last year and change).  He’s no Jackie Robinson in terms of talent.  That GM might consider that handicapping his team (that’s admittedly harsh, I know) to make a point doesn’t make economic sense, not to mention basketball sense.  And they’d be forgiven for that line of reasoning.

I hope that Collins’ decision paves the way for more players to come out.  We all know they are out there, and I hope that the fear of being first was all that was holding others back.  But at the core of the issue, I want to know what the character of the sports world really is.  I want to know that there isn’t a dark, discriminatory heart at the center of these institutions that, rightly or wrongly, influence so many young people around the world, and serve as a role model of supposed meritocracy.  I want to know there is good at the heart of sports, but on this issue, it’s impossible to know.  We need more athletes like Jason Collins. We need them now.

 

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Making as a Manager and Managing as a Maker

A while back, I read a post by Paul Graham about the differences between the manager’s schedule and the maker’s schedule.  It’s worth a read for anyone who works in an environment where technical people (who tend to be makers in this sense) and business people (who tend to be managers) interact.

I think the overall message is on the money - people whose job requires doing the work need long, uninterrupted periods of time to focus on hard tasks.  People whose jobs require keeping track of initiatives, managing progress to a timeline and budget, getting buy-in from stakeholders, and generally steering the ship need short, pointed opportunities (meetings) to accomplish these goals.  Different objectives, different modes of time management.  I also agree that these two styles of time management can cause conflict when they have to work together.  Mid-morning or mid-afternoon meetings won’t make the manager think twice, but can blow a whole day for a maker. 

Here’s where I start to diverge from the general ideas in that article: I’m a manager - by training, by job requirements.  I’m also a maker - by preferred working style, by other job requirements (I have a many-faceted job).  My schedule doesn’t fit nicely into either the maker or the manager paradigm.  Further complicating this issue is the fact that when I wear my manager hat, the projects I need to manage involve makers, so my management style needs to accommodate the maker schedule.  What’s the solution?

The way I’ve managed this problem in the past has been two-fold.  First, I work earlier than everyone else.  Coming into the office early in the morning affords me a long, uninterrupted block of time to work on my maker tasks (you may notice all of my blog posts are timestamped before 10am; that’s not an accident - most of my code is also written before 10am).  It also allows me to shift to the manager schedule in the afternoon without blowing the whole day.  Second, because I’m able to get work done early, I can usually block off some mid-day time to recharge and transition.  Going directly from maker mode into manager mode is hard.  Respecting the lunch hour and doing some mindless tasks (catching up on email, reading news, etc) makes the transition smoother.  It can cause some optical problems - despite all of the evidence to the contrary, the 8-6 (9-5 if you work in some industries, 10-4 if you work in government) work schedule is still held as sacred.  Taking a mid-day break to recharge runs the risk of making some people perceive me as lazy.  They are wrong, but it’s hard to manage the perception when “10 stressed hours of half-focus” is the norm.    

I don’t want to give up my maker side.  As I mentioned above, I prefer the maker style of work, but my career trajectory is heading towards an increasingly managerial role.  As such, getting the balance right is a high priority for me.  Any suggestions?

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An Open Letter to All Business School Professors

Dear Professor X,

You should stop using powerpoint for your lectures.  I know that every other business school professor is doing it too, but trust me, you need to stop.  Your slides, in the nice branded University format, are terrible.  They shouldn’t be projected on a screen.  They were designed for lazy students who don’t want to think, but just want to read the slides, nod their heads, and get back on Facebook.  Do you see the open laptops all around the room?  There is a conversation on Facebook about where to go for drinks after class.  You could have prevented this by not using slides in the first place. 

If you insist on using slides, I recommend that you watch 100 episodes of The Daily Show.  Does that sounds like a strange recommendation?  The images behind Jon Stewart are slides.  Did you realize that?  That is what your slides should look like.  They should be images with a few words, maybe a headline, that captures my attention.  As soon as you put a single bullet point on your slide, you have failed me.

Some of the people in your class don’t care.  They love your slides because they makes it easier to cram for your exam.  They will never think about your subject again, so they are happy you have chosen to make their lives easier.  I care.  I want you to teach me something that I’ll remember two years from now when a coworker asks me a question and the Very Important Thing you are talking about is relevant to the answer.   I don’t want you to make my life easier now.  I want you to make me better.  Making me better is hard.  Your slides are inadequate.

Teaching without your slides is harder too.  You have to remember exactly what you want to talk about.  You have to think on your feet.  It’s been a long time since you learned the basic Basics that you are teaching us about this subject, so you will struggle to communicate these concepts in terms that make me understand.  Your slides are a crutch.  Get rid of them - it will make you better too.  You do care about being a good teacher, right?  Oh, no. The University only cares about your research.  Teaching is at best a tertiary consideration.  It won’t help you get tenure, and if you already have tenure, then you really don’t need to worry about it.  I took a class called Economics of Incentives and I’d remember that the concepts were relevant here, but that professor used slides.

Try this for your next class.  Just ask a question, a really hard question, at the start of class.  Then don’t tell us the answer.  Make us ask you for more information - make us think.  Help guide us in the right direction, but occasionally mislead us by giving us some irrelevant information.  Have you heard of Socrates?  I know philosophy isn’t a business school subject, but he was a pretty smart guy.  He figured teaching out a long time ago.  Powerpoint hadn’t been invented yet, which probably helped. 

The projector in this classroom was made in the 1980s.  The computer still has a floppy disk drive.  Some of your students (especially your undergrads) don’t even know what a floppy disk is.  They are confused by the fact that to save a file you click on the blue letter “H.”  Most of your students, including your MBAs, don’t realize that we only call slides “slides” because they used to be in a slide projector.  We all saw the first season of Mad Men, but we haven’t put the pieces together.  Synthesis isn’t our strong point - we’re MBAs, not Chemists.  The point is that your technology is outdated.  If you insist on using technology, find a better way.  If I wanted to read slides, barely listen to you, and surf Facebook, I would take this class on Coursera. 

There are others like me in your class, students who are begging to be taught, students who want to learn.  Stop catering to the lazy people who want a credential without doing any work.  Teach to us, the ones who care, the ones who will appreciate the difficulty of the task of making us better.  I know you would enjoy teaching more if you could see us, your students, improve.  Let us make teaching more fun for you.  Make us better, and we will make teaching more fulfilling.  Let’s do this together. 

Now turn off the projector.

Sincerely,

—The guy in the front row

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I’ve had a few great professors in my MBA.  Among the best, only one used powerpoint slides while lecturing, and his slides were noticeably different.  I’ve had a few bad professors too.  Among them, only one didn’t use slides (he was actually the worst professor I’ve ever had, slides or no slides).  We’ll call those two, the great professor with slides and the terrible one without, the exceptions that prove the rule (though that’s a phrase I don’t understand. Maybe it was taught to me on a slide).  The general pattern holds.  Slides = bad.

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