High School Middle School Statistics

Estimation and Anchoring

A recent post by my Stats-teacher friend Anthony, “Wisdom of the Crowd“, reminded me of an estimation activity I have used many times in my 9th grade Stats class.  The activity is based on a chapter from John Allen Paulos’ book A Mathematician Reads the Newspaper.

You’ll need two groups of students; 2 different classes will do.  Each student uses an index card or a scrap of paper to write responses to 2 survey questions. I warn the students beforehand that the questions may seem strange: just do your best to answer as best you can.

  • Question 1: Do you believe the population of Argentina is MORE or LESS than 10 million people?
  • Question 2: Estimate the population of Argentina.

Allow a few moments between the questions for the inevitable blank stares and mumbling.  Then collect the responses.

For the second group, you will ask the same two questions, except that the first question will replace 10 million with 50 million.  After you have data from both groups, write it on the board or print it and hand it out. It’s time to analyze and compare. Challenge students to communicate thoughts about center and spread. Also, which group’s data do they feel does a better job of estimating question 2?  It’s a neat activity, and while you will receive some strange responses as estimates, and students will generally guess higher on question 2 if they have been anchored to the 50 million number.  Some guidelines for this activity are avilable.  Have fun!

According to Google, the actual population of Argentina is around 41 million.



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?