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!

First Day Plan: Calculus I

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.

The Art of Paper Folding: Kawasaki's Rose Bud

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...

The Mathematics of Games and Gambling - Planning Part 1

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!

Welcome!

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!

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