The Mathematics of Games and Gambling: A Course Update

Friday, September 28, 2012

I haven't done well keeping the blog updated on a daily basis (or even a semi-daily basis) but that doesn't mean that I haven't been keeping track of what has (and hasn't) worked in my classroom.  I've talked about my Mathematics of Games and Gambling course that I'm developing/teaching during the Fall 2012 semester a few times already (most notably here and here).  Since I haven't mentioned much about the course since then, it's probably time to reflect on how things have gone through (almost) five weeks.

So far in class (through five weeks - aka fifteen 65 minute classes), we have:

  • Learned the game of Chuck-a-Luck
  • Learned the game of Roulette (both American and European)
  • Learned the game of Craps (both street and casino)
  • Learned the game of Five Card Stud Poker
  • Learned about expected values
  • Learned about probabilities
  • Learned about choice trees
  • Learned about combinations and permutations
  • plus a variety of other (smaller) topics such as a few brain teasers and card puzzles, plus assorted vocabulary as it arises in various gaming contexts.

We've also had a day where the class played craps, another where they played roulette, and today they happen to have a Five Card Stud Poker tournament.  The class has had five take home assignments so far (all but one spanning multiple classes).  Finally, the class took their first test on Monday of this week.

With all that's been done so far - and, quite honestly, that is a LOT of material for a distribution math class, it's time to reflect on the good, the bad, and the things that I might (or might not) change next time I teach the course.
*Note:  All observations are mine and mine alone.  I have not polled my students yet nor issued any sort of survey thus far in the semester.
Image source: http://singlemindedwomen.com/women-relationships/duchess-digest-the-good-the-bad-and-the-solution/

The Good:
Students love games and they seem to be working to understand the mathematics as a way to get better at the games.  I split the class into two separate tables when playing Craps - one table had every student (except one) turn their $300 into $1000+, while at the other table almost all the students went bankrupt.  It was a great, though unplanned, lesson in the draw of the casino!

The first round of testing had 6 of the 18 students earn an A on the exam.  For a distribution math course, that's awesome.  Even more impressive is the fact that the test was five pages long - and certainly not easy - and yet one-third of the class earned an A.

The best part of the class though isn't the grades or even the "fun" stuff.  Instead, the best part so far has been the students reaction to various problems, homework assignments, and in class activities.  The overall attention and engagement levels are through the roof as compared to a typical distribution class.  I can't say the class is at 100% in terms of engagement, but on any given day I would wager that at least 16 of the 18 students are fully engaged.  Again, a remarkable number considering the fact that most of the students in the class are self-professed math-phobes.

The Bad:
I mentioned the results of the first exam in "The Good" section, but I would be remiss if I didn't mention it in "The Bad" section as well.  Out of the 18 students, 4 of them ended up with Fs, including one student who left a pair of 21 point problems (out of 100 total exam points) blank.  As I said above, the engagement is there for most, but not all, students.

I think it's a bad thing that I have been unable to fit in time to play most of the games with the students and let them figure out their own strategies before we go over the mathematical analysis of each game.  I was able to do that for Roulette but not for Craps.

Things to think about changing:
I don't think I have the exact right balance of rigor and exploration.  I have a tendency to automatically analyze games logically as soon as I am introduced to them (my wife hates that, especially if we are playing against each other)!  However, just because that's a natural thought process for me, it most definitely is not for the majority of the students in the class.  I need to be better at guiding them through the analytical process.  In the beginning of the semester, I used a few brainteasers/puzzles as warm-ups which worked quite well.  Unfortunately, I haven't done much of that lately (mostly because the amount of material in the daily lessons hasn't allowed extra time).

I think the first exam should have been in week 4 rather than week 5.  I probably should have spent a full day on permutations and then a second day on combinations.  I ended up doing both on the same day and while it worked for 80% of the students, that's not close enough to 100% for my tastes.

Finally, I need to come up with a way to assess learning besides exams.  Many of the students in the class that didn't score great on the exam actually seem to know the material but they "froze" during the exam.  I think old math paranoia habits die hard...

I guess there's not really much purpose to this blog entry other than as a self-diary of sorts.  Of course, if you have any ideas for the course, by all means share them!

The Job Application Process: I Didn't Miss This One Bit.

Tuesday, September 18, 2012

I've been teaching at my current institution on a full-time basis for about five years now.  Unfortunately for me, the college has a policy that people without a Ph.D. get canned after a set amount of time.  That amount of time seems to depend on who you talk to, some say 5 years (obviously not true since I'm still there), most say 6 years, and a few say 7 years.  Even in a "best case" scenario, I'd only be able to work there for about two more years.  As such, I have officially jumped back into the proverbial hiring pool.

My wife's job doesn't allow her to move from our current location so I'm bound geographically.  Unfortunately, for a person in higher ed, that's dangerous since there simply aren't that many jobs out there.  Of course, my options are further limited by my limited degree (and don't get me started on my mostly stellar teaching evaluations).

Therefore, despite still having a job for the moment, I decided I better start applying for any (and all) jobs that might be a fit for me if they are at all geographically close.  Last night, I finished an application for a vice president of student life position.  Sure, it isn't a teaching gig but it is still in higher education and I would still be able to work with students, two things that I find important.

Will I get the job?  Eh, I don't have any idea.  If I do, I'm sure the tone (and direction) of this blog will change.  Of course, I have about two readers per day so I don't think anyone will mind.

Until then, I'm still working and teaching...which means lots of lessons (and some grading).  In my Calculus class, we looked at the product and quotient rules for derivatives on Monday.  Tomorrow we will spend the class reviewing, though most of the review will be in the form of matching algebraic expressions (specifically with fractional exponents since incoming students to my calculus class seem to be universally weak with exponents/roots).

So I Played Pong in Math Class Today

Wednesday, September 12, 2012

That's right, I played pong in my Calculus I class today (assuming today is this past Monday).  No, it wasn't this kind of pong...
Image source:   http://www.aboutdwi.com/blog/tag/teenagers/ 

...but it was based on that idea - with my own "twist" of course.  

My class had their first exam (limits - I'm still doing Calculus I in the "conventional" order).  Anyhow, I often try to do some sort of review game with the class the day before the exam.  During my couple of years of teaching, the review game was almost always Jeopardy.  The students loved it then (they even added me to the "Men of Mathematics" poster in the hallway as the inventor of math jeopardy)  Fun times for them, but honestly, Jeopardy is a total bore from my perspective...especially when done over and over.

A couple of years ago, I managed to create a working "Who Wants to Be a Millionaire" game, complete with the lifeline allowing them to phone a friend using their cell phones.  Great hilarity ensues when a team of four or five is all on the cell phone trying to explain (and then acquire) an answer to a question within the 60 second time limit.

Flash forward to last year, I added in a game based on the bar trivia game (referred to as Quizzo in my neck of the woods).  I guess the bar theme got extended a bit further this year with my Limit Pong game.

Set-up:

I found four identical boxes (I used baseball card boxes - something I have plenty of) and labeled them as Team 1, Team 2, Team 3, and Team 4.  I then took a bunch of styrofoam cups (approximately 30) and used a sharpie to put a point value on the inside of the cup.  The point values varied, the majority of the cups were worth either +1 or +2, though I scattered in a few +5s and one +10.  To keep things interesting (and to add some strategy to the game), I also added in a few -1s and -2s on the cups.

From there, all I needed was a ping pong ball (I happen to advise the table tennis club on campus as well so that was easy enough to acquire).  For the first time, I had to spend a bit of time going over the rules, but I imagine if I play the game again with the same class I could easily fit in a fifth round (and stay within the 65 minute class period).

Rules:

The rules were fairly simple.  I split my class into four equal teams.  Each team member got a worksheet packet for his or her self AND the team got an extra "team packet".  I gave the teams a few minutes to work through the first page of the packet.  The goal of the problem solving round is to fill out the team packet with the team's final answers.  After time was up, the groups swapped packets and graded each other's pages.  At this point, I posted the answers on a PowerPoint slide so that students could copy the solutions on their own, personal packets if they wished.

Each round was worth a set amount of points (usually 1 point per question).  At the end of the scoring phase, one member from each team (this role rotated each round) came up to the front of the class as the "thrower" for that round.  Using the points earned during the round as a currency of sorts, the thrower could attempt to throw the ping pong ball (with a mandatory bounce) into their team's box OR they could select a cup (all cups were facedown so it was a mystery as to the point value) and then place the selected cup in a box.  Positive point cups go in their team's box, negative point cups get to go in one of the other team's boxes.  In my cases, each box could hold up to five cups, so once a box reached five cups you could stack (and therefore, replace less desirable cups).

We repeated the process three more times, with a different student getting the opportunity to throw each round.  At the end of the day, the highest point total (from the throws only) won!  

For the first attempt, I liked the game a lot (and it seemed popular with the students).

The good:

I did this activity in both of my Calculus I classes.  In one class, the students recognized there was a strategy in terms of whether to throw the ball in the hopes of scoring points and grabbing cups (in hopes of making future throws worth more).  By the way, a throw that lands in the box (but not a cup) was worth a single point.  You might even say the teams attempted to optimize their score by carefully choosing throws and cups.

The bad:

The other class didn't grasp the strategy at all.  The initial throwers all opted to fill their box with cups (as much as they could).  The next group also grabbed cups (replacing as necessary).  By the time the third and fourth throwers were up, they only had the option of throwing.  On the other hand, the average team score was much higher with this group - though rumor has it some of them were well-versed in the art of beer pong.

Things to try:

I need to emphasize the solutions a bit better.  Students got too excited about the throwing part and would sometimes not worry about problems they got incorrect.  I suppose that's the danger of any game where the students are emotionally invested, but it's still a problem that I need to fix for next time.  After all, what's the point of reviewing if the students don't make the best of the time and opportunity?

Game modifications:

The game itself worked fine, though I think it might be fun to have a bit more variety in terms of the number of cups.  Scores would have been more impressive if the boxes held more cups - some teams scored zero points simply because they were lousy pong players.  I don't mind skill having a role, but it didn't seem right that teams couldn't land a single ball in the box.  The students really enjoyed it when they grabbed a negative cup and got to place it in one of the opponent's boxes.  Both classes used some strategy in terms of the negative cups (using them to either wipe out big positive gain cups or to try and weaken the first place team).

All told, it was a fun activity that my students really seemed to enjoy.  I haven't finished grading the exams yet, so I can't even begin to make any guesses as to the effectiveness of the activity compared to previous year's games but I'm sure I'll try to make some conclusions at some point in the future.  Until then, if you have any review games or other ideas to share with me, please do so!

Week 1 is in the Books!

Tuesday, September 4, 2012


Happy (day after) Labor Day!  While I’m guessing most of you had yesterday off, my school held classes.  I’ve worked there for five or six years now so I’m used to working on Labor Day (and honestly, it’s hard to complain about working on the holiday when I just had roughly four months “off”).  No one feels sorry for me.

One of my goals for the blog is to keep a running diary of thoughts and observations from my classes.  After one full week, I have to admit that I’m looking forward to the semester!  I have a pair of sections of Calculus I, the first class has 16 students and the second has 19 students.  I appreciate having two sections with roughly the same number of students as it makes planning activities easier!  My first day of theCalculus course went extremely well, but I do fear that I’ve lost a little steam and enthusiasm from the students now.  I attribute that to a string of mostly lectures and problem solving in regards to limits.  If there is one giant weakness to my teaching, it’s that I don’t have a steady stream of varied activities.  Each semester I seem to plug in one or two new things to try (and I usually keep them for future semesters) but even so the course can get a bit monotonous. 

Historically, about half of my students fail to fully grasp limits (and continuity) by the first exam.  Most of the students are able to pass the exam, but limits at the Calculus I level are easy enough that I believe all the students should be able to muster at least a B on the exam.  So far, my current crop of students is right on pace – most have grasped the concept but there are still four or five in each class that still don’t understand.  I keep thinking that if I could come up with an activity that demonstrates the concept of the limit those students might also find themselves understanding all the course material.  Unfortunately, I have yet to come across a suitable activity.

My third class is the Mathematics of Games and Gambling – a course that I am currently developing from scratch.  I had ambition plans for the course but it didn’t even take a full week for me to realize that I am going to have to go quite a bit slower than I had initially planned.  After a week and a day, we have covered finite probabilities and played both Chuck-a-Luck and Roulette.  In the next class, we do a mathematical analysis of Roulette (other than explaining the rules, I didn’t say much about Roulette in class yesterday).  My goal was for the students to experiment as they played to see if they could come up with a strategy that was either good or bad. 

In terms of bad strategies, the students offered some advice like “don’t bet a dollar on black and a second dollar on red at the same time” and “don’t bet on single numbers.”  The first piece of advice is perfectly valid – and quite honestly, the second was as well in the sense that I only gave each student $10 to “bet” with.  Of course, we all know that betting strategies in roulette are about as helpful as a beach volleyball in the arctic circle.

All things considered, I’m moderately pleased with the first week of classes.  I wish I could figure out a way to teach Calculus without resorting to as many lectures but otherwise things have gone quite well.  I should have a better handle on my Gambling students’ abilities once I grade their first homework assignment (which was collected yesterday). I think I've managed to display most of the traits that my students consider important in a good mathematics teacher, though I don't think most of them find me funny*.

*When explaining the floor and ceiling functions, I like to ask my students what the floor of pi is.  After getting the correct answer, I point out to them that they now know the floor of pi(e) is not actually crust but three.  Each class produced three of four groans, a chuckle or two, and a lot of eyes (and heads) rolling.  

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