Category: Statistics

Categories
Statistics

Take-Aways From Kansas City

Last week, I had the opportunity to participate in the Advanced Placement Statistics reading in Kansas City  Over 600 educators convened to grade over 900,000 questions.  Heading into the week, I wasn’t quite sure what I was getting myself into, but the Facebook groups dedicated to both readers and us newbies (“acorns”) were extremely helpful and generated buzz for the week.  Also, the e-mails from Chief Reader Alan Rossman were helpful in getting us new folk acclimated.  So, what is a week as an AP reader like?

  • You’ll do a LOT of grading.  I spent 4 days straight grading the same hypothesis test over and over.  On other days, I graded the investiagtional task question.  By my count, I graded a little short of 2,000 papers.
  • The training and supports are outstanding.  My partner, Cora, was fantasitc and my table partners kept me from going bonkers by Saturday.
  • The organization requiredto move 150,000 test booklets efficiently around a conventional hall leaves me speechless.

But more than anything else, I meet dozens of people who share my same strange excitement over teaching Statistics.  To one colleague, I described the experience as hanging out with the “Stats all-star team”.  Everywhere you turned, there was an educator whose materials and ideas you had used and shared, and the opportunity to touch base with so many of them was fantastic.  (aside: I think an opening exercise for all stats classes should be to utilize an activity authored by Floyd Bullard.  Then, after the activity, have your students describe what they believe Floyd looks like.  Then, sit back as you show them what Floyd really looks like.)

My take-away from the week applies not only to Stats, but to all teachers I coach and classes I encounter.  Specifically, how can we utilize vetted performance tasks (like AP Stats questions) in classrooms as formative assessments?  Jason Molesky has done a fantastic job with this through his “Frappys” on the iconic stats website Stats Monkey.  The group of teachers I hung out with during the week are interested in using these problems and sharing papers and ideas online.  But why not do this in more classes?  What sort of problems are we giving in Geometry and Algebra which require students to reflect upon multiple standards?  How often do we ask students to peer-assess their work, and train them to look at external rubrics?

The best stats teachers I know utilize past free-response problems throughout the course, and use student work to forward instruction.  Wouldn’t it be great to have these resources for non-AP courses?  My dream here would be to work with a group of teachers to build a bank of tasks for Algebra I, Geometry and Algebra II, with rubrics to share.  I know many states have “free response” questions already, and we work with students in my home district to prepare for those.  But many of them are so clear-cut, so watered down in their expectation, that they are un-usuable as instructional tools.  Seeing so many alternate and valid approaches to problems at last week’s reading speaks to the quality of the questions being asked.  If you have any thoughts on written performance tasks, or want to work together on writing, shoot me a message here. 

Thanks to all of my new friends from the reading for making the grade-fest an event I will look forward to next year, and to Brian, Andy, John, Nick and Dave for the great conversations!

 
Royals game

Categories
Algebra Statistics

What’s the Probability That Quadratic Will Factor?

A comment from my post last week about the need for factoring led me to re-visit a question I have posed to classes before, but never allowed to move beyond the “gee, that’s interesting” stage.

Given a polynomial in standard form, with random non-zero* integer parameters a, b and c, what is the probability that the polynomial will factor?

I’ve pursued this question with classes before by writing a polynomial on the board, with blanks or boxes in the a-b-c positions.  Sometimes, I would take “random” shout-outs from the class to fill in the boxes.  With another class, the randint function on a TI calculator was used to generate our abc’s.  The point was to demonstrate that a large majority of quadratics are not factorable, and that despite the nice, rigged, problems we encounter in textbooks, we should spend far more time considering what to do with the messy ones.  But I’d never put pencil to paper and thought about the theoretical probability.

After my post on factoring last week, Jim Doherty mentioned a speaker he had encountered find an experimental probability that a quadratic would factor, and cited 7%.  That number seemed reasonable to me, but perhaps a bit on the high side.  I set up an Excel document to generate three non-zero integers (more on this later), and rigged a system to check for perfect-square discriminants.  I recorded experimental results, in groups of 1000 trials, and kept a running total.

Excel document

Quadratic Graph

After 25,000 trials, I found that 7.26% of the quadratics would factor.

*While this endeavor started off innocently and quickly enough, I had to start over after I realized my Excel document allowed for zeroes.  It took a little logical Excel rigging to exclude them.

So, there must be a theoretical probability out there someplace?  Anyone know how to do it?

Categories
Statistics

NCTM – Saturday

Much quieter here today, as the math folk move out and an antiques show moves in.  But still time for some sessions!

Essential Understandings in Grades 9 – 12 Statistics: Preparing for the Common Core.

Very excited for this session today, as one of the speakers is Roxy Peck, former Chief Reader of the AP Stats exam.  Also, looking for some ideas for our freshman year prob/stat course.

Big ideas –

1.  Data consists of structure + variability: look at math models but explore the big picture

2.  Hypothesis tests answer the question, “do I think this could have occurred by chance?” – what hypothesis is more plausible?

3.  To evaluate an estimator, you need to consider bias and the sampling method.

4.  Describe variability – distinguish different ways distributions are used (population, sample, sampling distributions) and be able to compare them. As more teachers are compelled to teach stats in HS, we need to train for the abstraction of sampling distributions.

How much do sample distributions tend to look like population distributions, and how can samples differ based on randomization?

5.  The way in which data are collected matters.  The is a risk of error that needs to be acknowledged and quantified.  Also, the collection method determines the type of inferential conclusions that can be made.  If the sample is not representative of the population, we should be suspicious of generalization.

“as statistics people, we are OK with being wrong 5% of the time” – Roxy Peck

All HS math teachers will find themselves in the dual role of being teachers of math and teachers of statistics. This is a bit scary to me.  While many resources are coming out which will assist teachers in presenting statistical ideas, I wonder how many math teachers are prepared to facilitate a discussion, perhaps over many class days, centering on one scenario and its many statistical concepts.  A colleague, at the end of the session ,offered that his department is “petrified” of the prospect of being made to teach stats courses.  Many great math teachers I know are like musicians playing classical music, adept and expert in the rhythm and complexity of math.  How many of my colleagues are prepared to become jazz musicians, and have conversations meander in new and exciting directions?

After attending 2 conference in the last month, and moving through meeting rooms, with various degrees of “fullness”, I have begin to develop the “Large Ballroom Theory”

Large ballroom theory:  given a large ballroom with many empty seats, people will

  • Find somebody familiar and sit with them
  • Sit in the back, for easy escapability from intolerable sessions
  • Sit in a location which will maximize the apparent fullness of the room, looking for bare spots

Philadelphia-20120428-00195, Uploaded by Photobucket Mobile for BlackBerry
It is the last bullet which I believe could be the start of a full-blown thesis.  In the photo above, think about where you would sit?  On the end?  Which row?  Someplace equidistant to others?  Or would you sidle up next to a stranger?