Categories
High School Statistics

My Favorite Teacher Circle: PASTA

Just got back from the fall meeting of my favorite local teacher circle, PASTA.  The Philadelphia-Area Statistics Teachers Association meets a few times each year to share best-practices in statistics teaching.  Many of this month’s presenters are AP Statistics readers, and the ideas are not specific only to stats…we just share great classroom action.  I gave a recap of our last meeting in the winter; enjoy the great ideas from our Fall meeting, and visit Beth Benzing’s website for materials from the meeting!

Daren Starnes, famous in the Stats-world as author of The Practice of Statistics, shared his first experience with Team Quizzes.  I have tried team quizzes before, mostly for quizzes where I knew students were having the most difficulties with material.  But Daren added some features I had not before considered:

  • Students are assigned to their teams at random.
  • Each team member received a copy of the quiz, and must complete the quiz.
  • In a quiz, one question is chosen randomly to be graded from each paper.  A student’s grade is a combination of the score they receive on the question, along with the average of the scores from the other papers in the team.

Daren also commented on the roles of introverts and extroverts in the teams, and how this method could empower introverted students to self-advocate.  He suggest the book Quiet: The Power of Introverts as a resource.

AdamAdam Shrager, famous as the social director and man-about-town at the AP readings, shared his movie-correlations activity.  This has become one of my favorite activities during the stats year.  Students are asked to fill out a movie-preference survey, which Adam then uses to compute peer-to-peer correltations in Excel.  (look for “correlation” in excel…you may need to activate the Stat Pack) Discussions regarding the interpretation of positive and negative correlations then occur.  Most importantly, mis-conceptions of the meaning of low or zero r-values are discussed with a context easily understood by students.

Table

Leigh Nataro shared her “Pacing a Normal Distance” activity, where students walked between 3 different campus buildings using “meter-long” steps.  The data is then entered into Fathom, and is used to discuss variability, the 68-95 rule, and normal probability plots.  Fun discussions of outliers and error as well!

Leigh

Our host, Beth Benzing from Strath Haven High School, shared a family income Fathom file which draws samples of various sizes from a clearly skewed distribution.  In addition to to having students record observations and work towards generalizations, Beth has worked to increase the rigor in her associated questions, using past AP items as her framework.  Some examples:

  • What is the probability that a sample of 5 families will have a combined income of over $500,000?
  • What is more likely: a sample of size 5 having a mean income of over $80,000, or a sample of size 25 having a mean income over $80,000?  You may recall a similar AP question from a few years ago regarding samples of fish.

Beth

Brian Forney shared ideas for bringing concepts from Sustainability to the AP Stats classroom.  In one example, Brian shared data on depths of ice sheets over time, with excellent opportunities to discuss cause and effect from scatterplots.  Check out Brian’s presentation on Beth’s website.

Finally, I was happy to share my recent lesson on Rock, Paper, Scissors and two-way tables.

The meeting concluded with some great ideas for making multiple-choice assessments more fair and effective.  There were a number of excellent ideas here, but I think I’ll look up some more info on alternate assessment methods and save it for another post…so stay tuned!

Categories
Geometry High School

A Math Teacher Ventures Into a Western Civ Class

During my prep period last week, I came back to my classroom after a trip to the main office and overheard some familiar language: Euclid, Pythagoras, The Elements.  What’s intriguing here is that my next-door neighbor isn’t a geometry teacher; rather, my colleague Glen is a social studies teacher, with 3 sections of Western Civ each day.  Excited, I popped my head into his classroom.  And after some good-batured ribbing out how he was advancing on my math turf, I went back to my prep.  But Glen and I later talked about our shared interest in the Greeks, which ended with an invitation to come into his class to share a brief math history lesson.  I’m no stranger to the occassional cross-curriculur lesson, so this represented a fun opportunity.

One on my favorite courses from my time at Muhlenberg College was “Landmarks in Greek Mathematics”, where I was fortunate to have William Dunham as a professor.  His enthusiasm for math storytelling has shaped my approach as a teacher, and his book, Journey Through Genius, was not only used in the course, but is a book I often come back to for inspiration and contextual reminders of math concepts.  The book both walks you through the mathematical landmarks (like Euclid’s proof of the infinitude of primes) and provides a backdrop of the places and people (like the fascinating battling Bernoulli brothers) which shaped the surrounding culture.  It’s a great resource for any math teacher.

For Glen’s classes, I chose an example which 11th graders could easily understand and which would provide a glimpse into the genius of the greek mathematicians: Eratosthenes’ approximation of the Earth’s circumference.

Eratosthenes observed that on the longest day of the year, sunlight would shine directly into a well, so that the bottom of the well could be seen.  But that farther from the well, in other towns, this did not occur.  The well was located on the town of Syene, which we now lies directly on the tropic of cancer.

Syene Well

In Alexandria, a known distance away from Syene, Erotosthenes measured the angle produced by the sun’s rays off a post in the ground.

SyeneAlex

Taking this further, we can use alternate-interior angles to use this same measured angle as one coming from the center of the earth.

earth

This central angle, along with the known distance from Syene to Alexandria, yielded an estimate of about 25,000 miles (or the Greek stadia equlivalent), an estiamate with an error of less than 1% of the actual circumference!  Both classes I visited seemed to enjoy this math diversion in the Western Civ class, with one student wanting to know more about how the Greeks approximated pi.

So find your local Social Studies teacher, and offer to bring in a little math!  There are some fascinating stories to tell.

Resources:

Excerpt from String, Straightedge and Shadow

From the Mathematical Association of America

From Jochen Albrecht, CUNY

Finally, from Carl Sagan’s landmark series “Cosmos”

Categories
Uncategorized

Exploring the MathTwitterBlogoSphere

explore MTBOSThis month, some of my Twitter Math Camp friends are hosting a fun, month-long event called “Explore the MathTwitterBlogoSphere”.  You can check out the website for more details, and each week promises a new task designed to encourage math teachers to reach out via blogs and twitter.

For the first weekly challenge, Sam Shah has asked participants to share their favorite rich task.  Even with having taught for 17 years, it was not easy to come up with one task which I felt summarized my philosophies, but here is what I feel is my best question.  It is one I have given many times in algebra 2, and our freshman-year prob/stat course:

How many zeroes are there at the end of 200! (200 factorial)?

That’s it.

Here’s why I like this problem, and why I enjoy giving it:

  • It’s has a simple premise.  Sometimes I need to embellish with “think about multiplying out 200!  It would be a really long number.  That number has a lot of zeroes at the end.  How many are there?”  But besides having to know what factorial does, it is plain and simple in premise.
  • It requires thinking about the nature of numbers.  Brute force doesn’t work well here.  When I first started giving this problem, I think I used 25 factorial, but then technology started to catch up with me.  One year, a few students used Excel, which gave a wrong answer, as it began to konk out at bigger numbers.  Even if students can now find an “answer” through some tech means, the challenge to explain the “why” remains.
  • The answer is secondary.  Communicating your reasoning is king.  This problem present great opportunities to utilize math vocabulary: factors, commutative property, grouping, etc.  I grade this task almost exclusively on communication, and students are often surprised to find that a math task can require such a level of revision and reflection.
  • I can move towards a generalization if I need to put my foot on the gas more.  If a few students seem to have the answer and communicate a solution, I can challenge them to develop a formula which works for any number factorialed (is this a word?).

Rich problem solving experiences have always been a part of my classroom culture.  This problem is one of my favorites.