## Saturday, December 22, 2012

Although one might take the title of this particular blog post to be an assessment of the craziness that is the end of another semester, I assure it's not meant to be that way.  Quite the opposite in fact.  My semester went quite well, especially the Mathematics of Games and Gambling class that I designed and taught for the first time.  As I've down throughout the semester (though admittedly not nearly as often as I would have liked), it's time for another update for the course (here's a link to my last few course updates...published back in September and another in October).

Throughout the semester, I covered a variety of topics in the class (as a reminder, the class was designed for non-mathematics majors to take as an elective to fulfill distribution requirements).  The topics that covered included (but are not limited to):
• Chuck-a-Luck
• Roulette
• Craps
• Keno
• Five Card Stud
• Texas Holdem
• Lotteries
• Instant Insanity
It is the final topic that shall be the focus of today's entry.

For a few semesters now, I've lobbied my mathematics department to allow me to teach an experimental course.  For the longest time, I wanted to do a 300-level Introduction to Graph Theory course.  Alas, our mathematics major is quite small (and most of our majors aim to teach at secondary schools so their electives get filled by education required classes like geometry).  As such, it has been deemed unlikely that I could ever get enough students to adequately fill up a Graph Theory course.

Last spring, I changed gears and proposed the Mathematics of Games and Gambling course instead.  As you can guess, that was approved (and has, for the record, been approved for the upcoming spring semester as well).  Although I couldn't teach graph theory, I still managed to find a way to squeeze in a few days of graph theory (at a basic level) by using the 1980s puzzle "Instant Insanity."

Before I describe how I taught the lesson, I will say that overall the lesson went extremely well.  I even put a question on the final exam which almost every student got correct!  If you have taught a lesson using Instant Insanity (or something similar), I'd love to hear about it.  If you happen to be inspired in some way to use what I did in your own classroom, I'd love to hear about your experiences when they happen!

Our class periods are just over an hour long, so for the first class of the unit I showed the students the original Instant Insanity advertisement after a short PowerPoint presentation that covered the various graph vocabulary that I wanted the students to know.

The terms weren't difficult, but they weren't all simplistic either (bipartite graphs).

The next class was much more of a riot (for me anyhow).  I gave each student a colored sheet Instant Insanity puzzle blocks which they had to cut-out and tape together.  Watching college students do something that many haven't done since a middle-school art class was definitely amusing (and somehow a bit worrisome too)!

After constructing the blocks, the students spent the remaining time trying to solve the puzzle.  While a few did succeed, the majority of the class did not successfully complete the confounding game.  I didn't provide any hints as they left class that day, though I did encourage them to find a "mathematical way to solve the puzzle."

The final day of the lesson was the big one - how to solve Instant Insanity puzzles using graph theory. To begin, we discussed how each block could be "unfolded" to a two dimensional image.

From there, we can easily pair the opposite sides.

Once the three pairs are made (front/back, top/bottom, left/right), all you need is a graph with four vertices (one for each of the four possible colors).  Draw an edge that for each of the three pairings.  For example, if the top color is blue and the bottom color is red, there should be an edge connecting the blue vertex to the red one.  If the left and right colors both happen to be yellow, there would be a loop connecting yellow to itself.  In total, there should be three edges for each graph (one graph per each of the four cubes).

Using the four graphs (or if you prefer, a single graph that combines the four small graphs), you then look for a path that uses an edge from each cube once and that makes each of the four vertices have degree two.  Once you find one such solution, find a second solution.  The first solution corresponds to the front and back of the four cubes.  The second solution corresponds to the left and right.   Note:  Once you've found the front/back and left/right, the top/bottom is forced (and therefore you do not need to worry yourself about those)!

I've skipped over a few of the finer details, but in essence that's how you solve an Instant Insanity puzzle.  For those who want to give it a go, here are three puzzles.  It should be noted that some puzzles may not be solvable.

Good luck!

## Saturday, October 27, 2012

Although I love almost all aspects of my job, there are a few things that I don’t necessarily look forward to doing.  One such thing is writing letters of recommendation.  Around this time of year, many of our undergraduate seniors are preparing for graduate school – and part of that preparation requires them to secure X numbers of letters of recommendation.

Don’t get me wrong, I love (most) of our seniors – and I definitely want to write them great letters.  I truly believe that they will do well in whatever program they choose, none of that is the issue at hand here.  No, the issue is all on me.  I’m a harsh critic of myself, and while you probably can’t tell based on the drivel that I put on this blog (among the two blogs that I maintain), I do care.

A lot.

Letters of recommendation are one of those things that can be super easy to write for a select student or two each year, but for the rest of the students…  Oh boy.  They can be exceedingly difficult, especially for students who I only had for a class or two or perhaps for a student who didn't get the best grades and/or perhaps didn't stand out in some other spectacular way.

For those of you who write letters for your students (or anyone for that matter), how do you do it?  How do you make the letters unique (or don’t you worry about that)?  Personally, I think a truly personalized letter is better – but then again, I've never served on an admissions committee.  I can’t say that I know for sure what they look for – and if I did, well, I’d guess my own letter writing skills would be a bit better!

## Monday, October 15, 2012

As I mentioned a few days ago, I took my Mathematics of Games and Gambling class to see a showing of The Price is Right LIVE at our local community arts center.  The show was last night and while I haven't seen my class yet to get their impressions of the show, here is my personal review of the Price is Right traveling show.

In a word:  Semi-lame.

Ok, that's not even a real word, but it's how I feel.

The good:

Many of people's favorite games are present.  Punch-a-Bunch, Plinko, Cliffhangers, Any Number, and Hole in One were all played.  The had one group of three people spin the big wheel.  Two people got the chance to play for the Showcase (only one showcase - both players bid at the same time).

The bad:

The show started late and ended early (or so it seemed).  Extremely short amount of time spent actually playing any games.  They showed a bunch of video clips from old shows - but nothing that you can't find on YouTube (for free).  The prizes were borderline good for most of the show...certainly not great (even with lowered expectations).  There was a refrigerator as a prize and a billiard table used as big prizes (neither were actually given away).  The small (initial bid) prizes included a 4 handheld phone system for a house (seriously, who uses those any more?), a pair of diamond earrings, a popcorn machine, and a vacuum.

The terrible:

The final showcase consisted of:
A new car (Nissan Versa I believe?)
A 3-day cruise to the Bahamas
An iPod touch
A 50 inch flat screen tv

The first contestant bid $19,000 and change. The second contestant bid$20,000 and change.

For the traveling show, the person who is closest to the actual price (without going over) wins ONE of the items in the showcase - in this case, the 3-day cruise.  In order to win ALL of the items, you had to be within $100 of the actual price (i.e. not going to happen)! Actual retail price (according to the show)?$14,000 and change.

The show ended on that note.  A seemingly bogus final showcase, a pair of losing contestants, and a bunch of audience members feeling like the contestants were cheating.  The best line I overheard while leaving the theater:  "If a new car really cost $14,000, then everyone would have a new car." Never mind all the other stuff in the showcase... Since I was curious, here's a few numbers that I found via internet research*: *note, all prices are guesses, I have no idea what the brands/companies were for some of the prizes Price of 2012 Nissan Versa starting at:$10,999 (from Nissan's website)
3-day Cruise:  $299 50 inch TV: LG ($699.00)  one of the cheapest options
iPod touch:  $179.00 Total:$12,176

So is the game rigged?  Well, I say yes but only because they make you think the showcase prizes are great when in reality they aren't nearly so good.  I also found it weird that in Punch-a-Bunch, the host knew exactly where the one $2500 prize was hiding...and it appeared to be printed on a larger card. Makes me wonder if that particular hole had two cards residing in it, a$50 or similar prize if the contestant happened to select it and the big prize otherwise (the host showed the big prize to "prove" the fairness of the game).  I say when you have to "prove" that you are on the up-and-up, you probably aren't really on the up-and-up.

My suggestion to anyone who might see one of the live shows in their area - bid $1 and nothing more on the final showcase. Chances are, your opponent will over bid and you then you win the cruise. Don't bother trying to get too close, it won't work! What I don't understand is why the producers of the show don't want to have one person win the cruise. If the price is really only$300, that's paid for in a matter of 10 balcony tickets...a mere drop in the bucket.  Why have people leave angry (even if the pricing seems fair now that I looked up all the costs)?  If one of the two people had won the cruise, I think the entire audience mood at the end of the show would have been much better.

I also question the length of the show.  In a typical TV episode (granted, I'm sure footage is cut in order to fit it in 60 minutes - with commercials), there are six games played, the big wheel is spun by two groups of three people, and there are two final showcases.

In the travel show, there were only five games played, the big wheel was spun by ONE trio of contestants, and there was only one final showcase.  The entire show lasted just over an hour - and much of that time was "wasted" by showing the aforementioned video footage of old shows and for people making their way to the stage.

In the end, I'm interested to hear my students' take on the show, but for me, I can't in good conscience recommend anyone go to the show.  You'll have more fun watching old clips on your computer - save the price of the ticket.

## Thursday, October 11, 2012

I've spent a fair bit of time discussing the happenings (and future plans) for my course that I am both developing and teaching this semester.  Today, I figured I would discuss our recent activity mostly to serve as another diary type entry for myself next semester.

My school's fall break is this week which means we get Friday off.  Yeah, a one day break isn't overly impressive (but we do get the Wednesday before Thanksgiving off as well which I appreciate).  Anyhow, it's often tough to cover much material in a short week for a distribution class since the students have all of Thursday, Friday, Saturday, and Sunday to forget.  However, thanks to our local community arts center, I got lucky.

You see, the traveling game show The Price is Right LIVE is coming to our town on Sunday.  When I learned about the show, I instantly arranged for tickets (free for the entire class) and rearranged my course schedule to accommodate the game show.  That meant both Monday and Wednesday were used to discuss Price is Right games and strategies.
 Image source: http://content.clearchannel.com/cc-common/mlib/2036/01/2036_1264503009.jpg

We spent one and a half days discussing the mathematics behind the game of Plinko!  It ended up being a great way to review choice trees (using a small version of the Plinko! board).  It also natural evolved into a way to discuss using a small problem to help us solve a larger version...and it perfectly illustrated the need for a formula or shortcut when trying to analyze the "real" board.  In our case, the short cut is of course Pascal's triangle...which in itself is an excellent lead in to the next segment of the course where we will discuss the Binomial Theorem (and the MLB World Series).

After spending the better part of two classes on Plinko!, I used the remaining class time to give a crash course on some other games that might be played during the live show (I emailed the producers of the program but they wouldn't divulge which games would be played).

We looked at when to spin and when to "hold" during the Showcase Showdown.

We talked about bidding strategies.

We talked about basic strategies for a few of the games...and we ended yesterday's class with a video of what not to do which drew the expected laughs.

Overall, the students seemed quite excited for the opportunity to attend the show.  The math wasn't easy for the Plinko! board but most students seemed to intently study the various problems and explanations.  I had plenty of great questions about it during the lessons which convinced me that the students were truly engaging the material.  I believe the allure of winning a car (supposedly they play a game where someone could win a car even in the traveling show) was a great motivator!

Now that the lessons are over, I can only sit back and hope at least one of my students get the opportunity to go the stage.  It'd be a lot of fun to say that students can take my class and possibly win a vacation to Bermuda or something!

## 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!

## 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).

## 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!

## 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. ### Qualities of a Good Math Teacher ## Tuesday, August 28, 2012 I had a highly successful first day (more on that another day perhaps). For today's post, I thought it would be fun to see what qualities my students thought a successful math teacher must possess. They answered this question (along with a host of others) within the first 10 minutes of the opening class. I have three classes to teach this semester: The Mathematics of Games and Gambling - a distribution level course. There is a mix of freshman through seniors in the course. Class size is 18. There is one math major in the course (a senior who needs credits). Calculus I - Section A - Mostly upperclassmen, about a third of which had me for a previous class. Class size is 21. Calculus II - Section B - All freshman (except two). Only one student had me for a previous class. Class size is currently 15 - it could potentially rise to 20 by the end of the week (though I doubt it). With that in mind, let's see what the students claimed as the qualities of a good math teacher (divided by class). Mathematics of Games and Gambling student responses: • Fun & Interesting • Thorough, patient • Interactive, knows the material, is excited about the material • Explains well, takes time, offers homework to help prepare for tests • Helpful & receptive to questions I may have; Makes subject matter interesting • Multiple ways to find a solution • Patient, eloquent, sense of fun • Explains lessons and hands on • Patient • Understanding, approachable • Knows what he is talking about; Is clear when explaining steps • Explains well and is hands on • Answers students' questions • Knows math; Gets the students interested in math (Texas Hold 'em? Yes please) • Strict, but fun, explains things, not everyone's BFF • Explanation skills, patience As you can see, there is quite a diversity in what makes for a good math teacher. However, the one word that appeared a lot is patience. By my count 7 of the 16 respondents either named patience outright or alluded to patience in their qualities. The other attribute that was named a lot was fun or excitement (four out of the 16 alluded to fun in their comments). For a distribution class, I thought the comments were quite telling! The students want to see why the subject of math is interesting or fun, but they want to be shown in a clear, precise, (and perhaps slow) manner. If I had to try to summarize this class' thoughts into a single sentence, it would be: A successful (distribution) math teacher has a love for the subject which they can translate into thoughtful, fun, and well-structured lessons. Moving out of the distribution course, I have a pair of Calculus I sections. First up, the (mostly) freshman section of Calculus and their thoughts on makes for a good math teacher. • Understanding of info, communication • Interactive, Hands on • Engaging, able to make course interesting and fun • Experience • They can somehow make class fun and not all lecture • Gives a lot of problems to practice with • Helpful, insightful, explains things well and goes over things extra times • Funny, helpful, smart • Can teach a topic in different ways to help better understand something • Organized, smart • Explains/teaches well, interactive, approachable • Enthusiasm For my (mostly) freshman section of Calculus I, the differences in what makes for a good teacher (as compared to the distribution course) were quite stark. The idea of being fun wasn't nearly as prevalent. The idea of patience was essentially non-existent. Instead, words like "smart", "insightful", and "organized" make an appearance. Since the majority of the students in the class hadn't had more than one college class (as in, a single meeting of one class since this class meets in the morning), it seems that many of the qualities are a direct response to the stereotypical college professor. Finally, my second section of Calculus I (a class in which there were only a couple of freshman and plenty of students who have had me for previous courses) had this to say about successful math teachers. • Able to fully answer questions & visibly demonstrate answer; able to make learning fun • Approachable • Explains everything • Patience, intelligence • Funny, respectful, patient • Someone who explains well and works problems on the board • Explains new material in detail • Goes at a good pace and is clear • Funny, helpful, interesting, knowledgeable • Patience, clarity, authority • Good at math • If they can teach me, they can teach anyone • Answers every question • Approachable • Interactive, entertaining, available • Doesn't assume we remember how to do everything from precalc* *not one of my former students for the record! I wasn't surprised by the across the board responses from this class based on the heterogeneous mix of students in the course. The freshman all said things to the freshman in the previous class while the upperclassmen usually went with something along the lines approachable or explains things well. Can you make any broad assumptions or observations from my data? I don't know (well, you can make assumptions but how accurate are they)? If you put all the responses into a single list, it's interesting how much higher "fun" and "patient" rank as compared to "knowledgeable". Even more telling, I teach at a liberal arts college so the answer of "prepares me for a future job" was essentially non-existent! Do you ask your students anything along the lines of what I did? Or, if you are a non-student reading this, what qualities do you think a good math teacher should exhibit? ### First Day of Classes ## Monday, August 27, 2012 It's finally here - the start of another school year! Today, I'll teach my first lesson in Calculus I (twice actually - I have two sections this semester). I will also teach the first class ever of The Mathematics of Games and Gambling at my college. Although I will have a busy semester, I am hoping to use this blog to chronicle my journey through the semester. Good luck to all of those who begin teaching today - and good luck to all the students as well! ### The Mathematics of Games and Gambling: Planning Part 2 ## Saturday, August 25, 2012 Note: This post originally appeared on my other (non-education related) blog Nachos Grande. I'm slowly moving all the education related posts over to this blog. Don't worry, there aren't many that need to be moved. This particular entry was the second in what was supposed to be a fairly lengthy series. Unfortunately, I never got around to writing a third planning post. Thus, any future planning posts you see in relation to my Mathematics of Games and Gambling class will be new!  Image source: http://entertainment.howstuffworks.com/how-to-play-roulette.htm This is the second in what will (hopefully) be an ongoing series throughout the summer as I plan my new course that I'm teaching this fall. You might want to read the first post in order to see why I'm doing this and to catch up with me. It's taken me over a month, but I finally have the first week of the course figured out and the lessons are (mostly) finished. When I teach, I usually use PowerPoint so part of my course planning involves making the PowerPoint presentation to go along with the day's topic. I also usually allow my students to print out the slides beforehand (if they so desire) so I have to add lots of animations to keep the work to problems (and the solutions) hidden from those who don't want to see the answers while they are working the problems out in class. Obviously, it is a lot of work - but based on student feedback, it's appreciated by the majority of them. Of course, there's always one or two students who tell me I should write on the board all the time instead of using the PowerPoint (they obviously haven't paid attention to my handwriting)! Before I put together the first week of lessons, I sketched out the full schedule of the semester - and if time allows (and there's enough interest out there), I'll probably do a post covering each of the major topics as I complete them. In this way, I can motivate myself to work harder all summer - and be accountable for my actions if I don't! I've decided to start the course with a week long study of finite probabilities. At the college where I work, the entire first week is known as Drop/Add week which means theoretically a student can add my class on Friday at 3:59 PM and expect to do well in the course (despite missing three classes already - it's a Monday, Wednesday, Friday morning class). As a professor, I hate the drop/add week because you can't cover too much or new students have little hope of catching up, but you can't cover too little because then the students who attend the first week get bored and think the class is lame. It's a tough life. So, I usually do some form of compromise. In this case, it's going to be a week of finite probabilities. We'll look at dice and cards on Monday and Wednesday and then switch over to Roulette on Friday. I think I can arrange to get a roulette wheel for the day so we'll probably "play" a little Roulette as a class as well. I'm debating about giving each student a "gambling bank account" of fake cash to use over the course of the semester. Will they listen to the odds and keep the money safely in the bank or will one student's success convince ten others to try their luck (with undoubtedly poor results)? It should be fascinating (at least for me). Once the second week begins, it's the proverbial pedal to the metal as we plow through a bunch of topics. I think the second week will pick up with Roulette and mathematical expectation. Speaking of that, did you know that the expected value of a$1 bet on ANY* type of Roulette bet on an American Roulette wheel is -0.0526?  That is, you should expect to lose on average 5 cents for every dollar you bet in Roulette.  No wonder the casinos love the game!

*There is one bet (the five-number bet) on the American wheel that gives the house an even greater edge of 7.89% or almost 8 cents per dollar.  See the image below for more details on the possible Roulette bets on an American wheel.

And for those that are wondering, there are two types of Roulette wheels in use around the world.  The "original" wheel (which is used basically everywhere except in the US) has the numbers 1 - 36 plus a green 0 spot.  The American wheel is identical except it adds a green 00 spot to go along with the single 0 spot.  In other words, the house advantage doubles on an American wheel!  It's not hard to see why Roulette isn't terribly popular in America and yet it is quite popular everywhere else around the world.
 Image source:  http://www.bestamericanroulette.com/roulette-guide/american-roulette-odds/

For the first week of my lecture, the main goal is for students to learn how to compute expected values (such as the one quoted above for Roulette).  Expected values aren't terribly difficult (but they prove to be an extremely valuable tool when analyzing some games)!  Essentially, you can compute expected values by summing up each individual event's probability multiplied by that event's respective payoff.  The one catch is that each of the events with probabilities have to be pairwise disjoint.

As with any course that discusses mathematical expectation, the section will conclude with a problem known as St. Petersburg paradox.  Essentially, the idea of the paradox is to flip a coin as many times as you can until you flip a tail (which forces you to stop).  The payoff is based on the number of heads you flipped before landing the tail.  If n represents the number of heads you landed, your payoff is 2^n.  The questions (that I won't answer here for those who want to think about it) are:

• What is the expected payoff for this game?
• Is it realistic?

If all goes well, I should be able to write about games like Craps and Chuck-a-Luck and maybe even some computational rules (along with counting rules) next time as that's where the course is now headed.  I've got about two more months of summer to make this happen!

## Friday, August 24, 2012

I hope to document some of my classes throughout the upcoming semester on the blog.  I figure it might be a good way to spark some dialogue with other readers on ideas and (especially) improvements.  It will also serve as a nice diary of sorts for myself when it comes time to assess what did and did not work.

For today's post, I present my plans for Day 1 of my Calculus I course.  In the past (I've taught Calculus I for three or four years now), I jumped right into the first lecture on limits after spending about 10 minutes going over the highlights of the syllabus (and explaining our course management software - Moodle).  After the introductory lesson I limits, I usually handed the students a PreCalculus (and algebra) review worksheet.   Here's one page of my review worksheet...

As you can see, the worksheet isn't trivial for the students - but it's terribly boring.  Even worse, there are three more pages of similar work, including a page of graphing.  I have found that my students need the review (hence the worksheet) but based on many of the edu-bloggers posts that I've read over the summer, I decided to try something different this year.

This time around, I have different ideas (most of which I've stolen from a variety of great bloggers).  I plan to greet my students as they arrive in the door with a PowerPoint displaying the course name (and my name).  That way, in case a poor freshman wanders into the wrong classroom he or she will (hopefully) figure it out right away and leave before they get too embarrassed by the gaffe.  As the students file in and admire my beautiful slide (just kidding), I will hand them a survey called "Who Am I?"  I stole the idea (even the name) from Dan over at dy/dan.

This will give the "early birds" something to do and I hope it will also set the tone that my class isn't going to be a "typical" math class (whatever typical means...mostly bad things I'm afraid).  After I introduce myself, we are going to split up and jump right into an ice breaker activity (cup stacking, also stolen from dy/dan).

At this point, I should be able to say that I've spoken to every student in the class and hopefully interacted (either formally with math questions or informally based on something I saw on their Who Am I? page).  Following the cup stacking activity (which I'm estimating about 10 minutes - I have a PowerPoint presentation that will serve as a prompt for each question).  The students will first guess how many cups are necessary by themselves and then answer all remaining questions as a group (including comparing each of the guesses).  I just went through my school's Writing Across the Curriculum training so the idea of writing responses to questions is fresh in my mind!

At this point, there should be about 30 minutes remaining (give or take a few) in a 65 minute class.  With most of the remaining time, I'm going to do a variation of a drawing activity (that I also saw on a blog but I don't remember where - probably dy/dan since that's where everything else came from but I can't be sure).  Basically, the idea is to have one student in each group act as the "eyes" and a second to act as the "hands".  The eyes are looking at a picture on the PowerPoint slide while the hands have their back to the screen.  The eyes have to describe what they see and the hands have to listen to the description and make the best drawing they can.

My pictures begin with a happy face (very simple) and then progress through an increasingly challenging series of graphs (linear all the way to horrible discontinuous graphs).  In fact, one of the graphs was pulled directly from my Day 3 lesson (foreshadowing - ooooh!).  The final drawing should be a riot - after a bunch of mathematical functions, the final drawing is of a train (and fairly detailed at that).  It should be fun!

Finally, in the last couple of remaining minutes in the class, my students will be given their assignment - essentially a longer survey (with a little bit of mathematical review thrown in).

I've scanned the final page of the survey, mostly because it's the one original bit (the beginning questions in the survey are inspired by a post on Long tails of \int e^r est) of my day one lesson.

As I write out my lesson (something I rarely do besides the PowerPoint), I think I've got a day that should be more interesting than the usual "syllabus day" for day one.  In fact, you might have noticed that I didn't mention going over the syllabus at all - I actually have three slides dedicated to it (one on my contact information and office hours, one on the grading scheme since that's all students care about anyway, and the final on the Moodle course page password).  The syllabus is wedged in between the two main activities - and will be handed out while students are reorganizing themselves.

I have no idea if the new plan will make the class any better at algebra and/or PreCalculus as compared to previous semesters where the main assignment was the lengthy worksheet, but if I had to guess I'd say the students won't be any worse on average.  I know for a fact that the day will be more fun - I hate syllabus days as much as anyone, and the students seem to hate it if I jump right into a lecture on day 1.  Hopefully they will be too busy having a good time to realize that they are learning (or at least reviewing) mathematical concepts.

"Did you just describe that graph with the word multiplicity?  What does that mean again?"

At least, that's what I hope to hear on Monday.  We shall see.

## Thursday, August 23, 2012

As I'm sure you'll see by reading more of my blog (or check my other blog to find bits and pieces of information scattered about), one of my non-sport interests (besides teaching math) is origami.  I happen to use origami in the classroom occasionally (especially during our math colloquiums) but that's not the point of this post.

Instead, I'd like to highlight one of the more famous "modern" models in orgami, Kawasaki's Rose.  According to Kawasaki's book Roses, Origami, & Math, the rose is an extremely challenging model to fold.  Since my wife asked me to fold three flowers to send to a friend in the Navy who lost a family member recently (real flowers wouldn't ship overseas nicely), I decided to do the next best thing as compared to the rose, I folded three of Kawasaki's Rose Buds.

The rose bud model is almost the same as the rose model - and as you can hopefully see via the instructions and photos below, the end result is still quite beautiful.

The first step is to crease the paper into 64 squares...

Then, do what Kawasaki coined as a twist fold.  Basically, the paper shrinks to a 7 by 7 grid with the overlap happening in the middle of each side.  The twist results in a small square being formed in the center of the paper.  With a little practice, this fold is actually fairly simple.
After doing the twist, you slowly gather up each of the four corners to form what sort of looks like a long rectangular box.
You can see the box shape here.  The right edge (in terms of the photo) is opened up slightly in order to form the first of the four petals.
For your first attempt, I highly recommend using small paperclips to hold the various folds together.  However, when all is said and done, the flower will stay together without any outside assistance.  I hadn't tried to fold the Rose Bud before so I used the paperclips for the first flower.  I did not use them for the second two attempts - each of which were quite a bit easier than the first try!
There's one finished Rose Bud...and now, three more...

## Tuesday, August 21, 2012

 Image source:  www.casinogames.org
*Note*  This post was originally published on my other blog.  I'm planning on moving my education related posts over to the new blog - what better way than to begin with the course that has inspired me to write this new blog?

By now, most of my loyal readers probably know that I teach college mathematics.  In fact, I’ve been teaching at the college level since the year I got out of undergrad (starting with a teaching assistantship in grad school).  I’ve been fortunate that I haven’t had to scramble (too much) since then in order to have a teaching job for each upcoming academic year.  For the past five years, I’ve been teaching at a four year undergraduate institution (which I love) teaching mainly Calculus and PreCalculus (though I’ve also dabbled in Combinatorics and Statistics a couple of times).  Well, I’m pleased to announce that next fall I’ll be teaching a brand new course titled “The Mathematics of Games and Gambling.”

Throughout the summer (as I plan the course), and perhaps throughout the school year as well, I plan to occasionally post a blog entry or two on my teaching progress along with any ideas/problems that I have with the course.  It is my hope that I’ll have a bit of “diary” of sorts in regards to the course.  For now, I’m thinking I’ll post the entries here at Nachos Grande but I may end up starting an “education only” blog just to keep things somewhat on theme at both blog destinations.

Luckily, for today’s post, there is a tie between one of the first topics I want to discuss in my course and the main ideas of the blog – namely, gambling!  In my course, we are going to mostly look at gambling in terms of casinos but we may also dabble in sports gambling a bit, specifically with odds and various ways to make (or lose) money.

The college that I work at is located in Pennsylvania, but we draw a lot of students from surrounding states (including a whole bunch students from New Jersey).  As such, current articles such as that discuss the legalization of sports betting in places like New Jersey will probably get a lot of play in the course.  As someone who tries to teach without always resorting to the “lecture and exam” pattern that so many math professors find themselves trapped in, I am always looking for new and exciting ways to draw my students in to the material and concepts.  Anytime you can draw in a student’s hometown (or at least home state), there is an instantaneous connection!

In fact, this particular course is designed entirely around the idea that interested and motivated students will learn mathematics!  The course isn’t for our majors (though I think there is a math major or two signed up “just for fun”), but rather, it is designed to give our distribution students a different outlet for their mathematics requirement (other than the usual statistics course).  Obviously I have no idea at this point whether or not the plan will work, but I can say that the course had strong numbers during spring registration which leads me to believe that I am on to something!

Anyhow, back to the structure of the course – one of the things that I’m debating about in my head (and now on the blog) is how much weight and importance to give various topics.  The course description is intentionally vague, and I do want to be sure to feature some games/activities that the individuals in the class find interesting, but there are other topics that I always want to make sure I include within the course.

There will be a few topics that are no brainers – poker for one since we get a fun way to experiment with ideas of permutations and combinations.  Other no brainer topics will include roulette, black jack, and Monopoly.  I’d like to include backgammon because there’s a lot of approachable mathematics in that game, but quite honestly, I’ve never played the game myself so I don’t know that I’d be the best instructor for that one!  I plan to look at games like Chuck-a-Luck, Bingo/Keeno, and Connect Four as well.  I’d also like to discuss and investigate some online casinos, like bwin casino (which also happens to be a huge sponsor for the Portuguese soccer league cup)!

Besides the common (and not-so-common) games in the course, I also want to spend a bit of time discussing gambling.  We will take a look at state (and multi-state) lotteries – and I have a fun little “Tips to win the Lottery” article that I printed off back when the gigantic MegaMillions lottery was all the rage.  That should draw some laughs (or at least I hope it will draw laughs by that point in the semester once students realize the follies in “hot numbers” or “numbers due”)!

I also plan to spend a little bit of time discussing sports (and prop) betting – specifically talking about betting lines and how they affect payouts (and betting practices)!  I have already spoken to one of the psychology professors about a guest lecture on the psychology of gambling as well – it should be a great, well-rounded course!  Who knows, it may even help to shape the future direction of other distribution mathematics courses, both at my current institution and at other colleges!

## Monday, August 20, 2012

Welcome to my newest blog:  Trial by Blogging.  It is my goal to use this blog to highlight my various attempts at improving education.  Some of you may know me from the baseball (and baseball card) centric blog:  Nachos Grande.  I won't be giving up that blog anytime soon but I have decided to try and split the education related posts away from the sports stuff.  Of course, I would encourage you to check out my other blog as well if sports are something you are interested in!

In order for you to have a little bit of information of where I am coming from, allow me to introduce myself.  I am an (almost) 30 year old mathematics instructor.  I've been teaching at various colleges almost non-stop since I graduated from my undergraduate college.  I attended a 4 year liberal arts college majoring in both mathematics and computer science.  Math ended up being my true love though so I did a bit of advanced schooling, eventually culminating in a Masters degree in education (with a math concentration).  It's a long story as to how I ended up with that particular degree, perhaps for another day.  Since obtaining my Masters, I've been teaching at the same four year college in the Mathematics department.  To date, I have taught PreCalculus (with labs), Calculus I (with labs), Introduction to Statistics, Combinatorics, and our majors colloquium.  I am also currently designing (and teaching) a brand new course offering:  The Mathematics of Games and Gambling.

You will probably find plenty of more background on myself in future blog posts.  For now, it's a matter of trying to find a template that I don't despise and getting ready for the new semester (which begins in six days)!

Finally, if you have your own education related blog, let me know in the comments below.  I'd be happy to add your blog to my blog roll!