Monthly Archives: August 2014

Probability and Monty Hall

I’m teaching AP Statistics for the first time this year, and my first week just ended.  I’ve taught statistics as portions of other secondary math courses and as a semester-long community college class, but never under the “AP” moniker.  The first week was a blast.  

To connect even the very beginning of the course to previous knowledge of all of my students, I decided to start the year with a probability unit.  For an early class activity, I played the classic Monte Hall game with the classes.  Some readers will recall the rules, but here they are just in case you don’t know them.  

  1. A contestant faces three closed doors.  Behind one is a new car. There is a goat behind each of the other two. 
  2. The contestant chooses one of the doors and announces her choice.  
  3. The game show host then opens one of the other two doors to reveal a goat.
  4. Now the contestant has a choice to make.  Should she
    1. Always stay with the door she initially chose, or
    2. Always change to the remaining unopened door, or
    3. Flip a coin to choose which door because the problem essentially has become a 50-50 chance of pure luck.

Historically, many people (including many very highly educated, degree flaunting PhDs) intuit the solution to be “pure luck”.  After all, don’t you have just two doors to choose from at the end?

In one class this week, I tried a few simulations before I posed the question about strategy.  In the other, I posed the question of strategy before any simulations.  In the end, very few students intuitively believed that staying was a good strategy, with the remainder more or less equally split between the “switch” and “pure luck” options.  I suspect the greater number of “switch” believers (and dearth of stays) may have been because of earlier exposure to the problem.  

I ran my class simulation this way:  

  • Students split into pairs (one class had a single group of 3).  
  • One student was the host and secretly recorded a door number.  
  • The class decided in advance to always follow the “shift strategy”.  [Ultimately, following either stay or switch is irrelevant, but having all groups follow the same strategy gives you the same data in the end.]
  • The contestant then chose a door, the host announced an open door, and the contestant switched doors.
  • The host then declared a win or loss bast on his initial door choice in step two.
  • Each group repeated this 10 times and reported their final number of wins to the entire class.
  • This accomplished a reasonably large number of trials from the entire class in a very short time via division of labor.  Because they chose the shift strategy, my two classes ultimately reported 58% and 68% winning percentages.  

Curiously, the class that had the 58% percentage had one group with just 1 win out of 10 and another winning only 4 of 10. It also had a group that reported winning 10 of 10.  Strange, but even with the low, unexpected probabilities, the long-run behavior from all groups still led to a plurality winning percentage for switching.

Here’s a verbatim explanation from one of my students written after class for why switching is the winning strategy.  It’s perhaps the cleanest reason I’ve ever heard.

The faster, logical explanation would be: if your strategy is staying, what’s your chance of winning?  You’d have to miraculously pick the money on the first shot, which is a 1/3 chance.  But if your strategy is switching, you’d have to pick a goat on the first shot.  Then that’s a 2/3 chance of winning.  In a sense, the fact that there are TWO goats actually can help you, which is counterintuitive on first glance. 

Engaging students hands-on in the experiment made for a phenomenal pair of classes and discussions. While many left still a bit disturbed that the answer wasn’t 50-50, this was a spectacular introduction to simulations, conditional probability, and cool conversations about the inevitability of streaks in chance events. 

For those who are interested, here’s another good YouTube demonstration & explanation.

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Math Play and New Beginnings

I’ve been thinking lots lately about the influence parents and teachers have on early numeracy habits in children.  And also about the saddeningly difficult or traumatic experiences far too many adults had in their math classes in school.  Among the many current problems in America’s educational systems, I present here one issue we can all change.  Whether you count yourself mathphobic or a mathophile, please read on for the difference that you can make for yourself and for young people right now, TODAY.

I believe my enthusiasm for what I teach has been one of the strongest, positive factors in whatever effectiveness I’ve had in the classroom.   It is part of my personality and therefore pretty easy for me to tap, but excitement is something everyone can generate, particularly in critical areas–academic or otherwise.  When something is important or interesting, we all get excited.

In a different direction, I’ve often been thoroughly dismayed by the American nonchalance to innumeracy.  I long ago lost count of the number of times in social or professional situations when parents or other other adults upon learning that I was a math teacher proclaimed “I was terrible at math,” or “I can’t even balance my own checkbook.”   I was further crushed by the sad number of times these utterances happened not just within earshot of young people, but by parents sitting around a table with their own children participating in the conversation!

What stuns me about these prideful or apologetic (I’m never sure which) and very public proclamations of innumeracy is that NOT A SINGLE ONE of these adults would ever dare to stand up in public and shout, “It’s OK.  I never learned how to read a book, either.  I was terrible at reading.”  Western culture has a deep respect for, reliance upon, and expectation of a broad and public literacy.  Why, then, do we accept broad proclamations of innumeracy as social badges of honor?  When an adult can’t read, we try to get help.  Why not the same of innumeracy?

I will be the first to admit that much of what happened in most math classrooms in the past (including those when I was a student) may have been suffocatingly dull, unhelpful, and discouraging.  Sadly, most of today’s math classrooms are no better.  Other countries have learned more from American research than have American teachers (one example here).  That said, there are MANY individual teachers and schools doing all they can to make a positive, determined, and deliberate change in how children experience and engage with mathematical ideas.

But in the words of the African proverb, “It takes a village to raise a child.”  Part of this comes from the energetic, determined, and resourceful teachers and schools who can and do make daily differences in the positive mindsets of children.  But it also will take every one of us to change the American acceptance of a culture of innumeracy.  And it starts with enthusiasm.  In the words of Jo Boaler,

When you are working with [any] child on math, be as enthusiastic as possible. This is hard if you have had bad mathematical experiences, but it is very important. Parents, especially mothers of young girls, should never, ever say, “I was hopeless at math!”  Research tells us that this is a very damaging message, especially for young girls. – p. 184, emphasis mine

Boaler’s entire book, What’s Math Got to Do With It? (click image for a link), but especially Chapter 8, is an absolute must-read for all parents, teachers, really any adult who has any interactions with school-age children.

Boaler

I suspect some (many?  most?) readers of this post will have had an unfortunate number of traumatic mathematical experiences in their lives, especially in school.  But it is never, ever too late to change your own mindset.  While the next excerpt is written toward parents, rephrase its beginning so that it applies to you or anyone else who interacts with young people.

There is no reason for any parent to be negative about the mathematics of early childhood as even the most mathphobic of parents would not have had negative experiences with math before school started.  And the birth of your own children could be the perfect opportunity to start all over again with mathematics, without the people who terrorized you the first time around.  I know a number of people who were traumatized by math in school but when they started learning it again as adults, they found it enjoyable and accessible. Parents of young children could make math an adult project, learning with their children or perhaps one step ahead of them each year. -p. 184

Here’s my simple message.  Be enthusiastic.  Encourage continual growth for all children in all areas (and help yourself grow along the way!).  Revel in patterns.  Make conjectures.  Explore. Discover.  Encourage questions.  Never be afraid of what you don’t know–use it as an opportunity for you and the children you know to grow.

I’ll end this with a couple quotes from Disney’s Meet the Robinsons.

Robinsons1

Robinsons2