Categories
Algebra Statistics

Developing Math “Spidey Sense”

A recent post in the math blog Divisible by 3 made me reflect upon the role of importance of estimates and initial gut feelings in math class.  In the blog post, Mr. Stadel shared some estimation anecdotes from his middle school classroom, and a great Ignite video on quality instruction from Steve Leinwand (do yourself a favor and watch Steve’s 5 minute rant on instruction….you’ll be glad you did!).  How often do we challenge our students to communicate initial guesses or predictions of what will happen next?

Recently, I tutored a young man named Kevin in AP Statistics.  Like many AP Stats students, Kevin was quite comfortable with using his calculator, to the point where I often grabbed the calculator from the table as he was reading a problem.  Consider the following problem:

In a recent survey, adults were randomly selected to provide their opinion on presidential campaign spending.  200 adults were randomly selected from Pennsylvania, and 200 were randomly selected in California.  In PA, 130 of the adults supported campaign spending limits, while only 122 in California supported limits.  Do the data show a significant difference in the opinions of all adults in the two states?

Before diving into the computations, I found it helpful to ask Kevin what his “spidey sense” told him about the problem?  Does the result “feel” significant?  Could he predict the p-value?  Does the student have a feel for what the numbers might bear out after we crunch them?  These “spidey sense” discussions were fruitful, in that the conversation would focus on important concepts like the effect of sample size on sampling distributions.  Have your students make initial predictions before performing any computations, and see how understanding and misconceptions are revealed.

spidey sense

How can we develop “spidey sense” in other high school courses?  Consider the following scenario, which is often used to introduce systems of equations:

  • Dave and Julie are each saving money in a bank account for a new television for their room.  Dave started his savings with $80, and adds $10 each week to the account.  Julie started with no savings, but adds $15 to the account each week.

I have used problems similar to this one, and often the lesson requires students to create a data table for each week, graph their data, and answer a series of questions which lead to Julie having more money than Dave.  Eventually, algebraic expressions are introduced and we can solve systems!  Ta-dah!

But since this is a money problem, there’s a great opportunity here to communicate and develop initial opinions.  Ask your class, “Who will be able to buy the TV first?”, “What does your spidey-sense tell you?” and see how many of the concepts develop organically.  Or, if you want to compare the two students, ask which of the following is true?

  1. Dave will always have more money than Julie.
  2. Julie’s savings will pass Dave’s soon.
  3. Julie’s savings will eventually pass Dave’s, but it will take a while.

Let the discussions drive the instruction.  Let students tap into their built-in intuitions and share ideas.  And, as Steve Leinwand exclaims, “value and celebrate alternative approaches”.

How do you challenge students to tap into their math-spidey-sense?

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
Uncategorized

Conic Sections Project – Part Deux

The conics project has produced the most hits of any post on the mathcoachblog to date (including over 25 from the Netherlands….I can’t explain it, but it’s pretty cool).  A few comments and e-mails to me requested a rubric for the project.  To be honest, I don’t have an uber-detailed rubric for this project beyond the file I share below…

http://dl.dropbox.com/u/68005919/Conic%20Sectins%20Drawing%20Rubric.doc

Kevin and I discussed some ideas and modifications to grading for this project, and I give you all these suggestions:

First, as a mentioned before, the peer grade can be achieved by giving all students 5 sticky stars. Have them discuss guidelines for choosing the best works beforehand, then allot their stars and stick them to their favorites. Also, you will notice that I am a bit of a stickler for having students leave their names off their works. This is intended to eliminate some bias from having students select their buddies. Also, if possible, have different sections of students grade other sections of student works.

Next, I would alter the requirement of having students provide all equations they used. Realistically, I never looked at all of the equations anyway. An interesting compromise would be to have students select 3 equations, and provide an explanation for the transformations and stretches needed, and how they play into the overall theme. This would be an improvement over a laundry list.