Category: Statistics

Categories
Statistics

Statistical Illiteracy, Education Policy, and Home Runs

Bob, they printed teacher salaries in the newspaper this week, and it said that the median salary for your school district is $65,000.  That’s great.  How long until you make $130,000?

Yep, that’s my mother.  A good cook, but a bit misguided statistically.  A recent CNN interview of Diane Ravitch reminded me of the need for statistical literacy, and the danger of inappropriate statistical conclusions.

It has been my intent to keep this blog non-political.  Of course, I have my opinions on educational issues at both the national and local levels, but my mission here has been to provide ideas and resources for my math teacher friends.  So now I will try to tip-toe through a recent CNN interview between Diane Ravitch, the former Assistant Secretary of Education, and CNN reporter Randi Kaye.  Diane expressed her concerns over statistical misuse in the interview through a recent blog post:

Randi Kaye asked me about NAEP scale scores, which was technically a very dumb question, and I was stunned.  She thinks that a scale score of 250 on a 500 point scale is a failing grade, but a scale score is not a grade at all.  It’s a trend line.  She asserted that the scale scores are a failing grade for the nation.

Fortunately, CNN chose to cut this portion from the aired interview.  But the statistical arguments are still worth exploring and learning from.  Consider how many parents would feel if their child were to score 600 on the math portion of the SAT.  What does this number represent, and how should it be applied?  On  Diane Ravitch’s blog,  this SAT example is used to relate to the misuse of the NAEP statistic:

That is like saying that someone who scores a 600 on the SAT is a C student, because it is only 75% of 800. But that’s wrong.  The scale is a technical measure. It is not a grade, period.

And while the SAT analogy helps in explaining the misuse of scale scores, I feel it only tells part of the stats abuse story.  The use of a group average to reach a conclusion about all members of the population is simply inappropriate.  Given that Finland is often cited as having the top mean NAEP scores, does this imply that all of their schools are passing?  Of course not, and it would be foolish for their schools to not work to improve.  If, instead of country mean scores, we plotted individual student scores, what would we see?  I suspect that we would see each country represented with sprinklings of students at the top, many students represented in the middle, and sprinklings towards the ends.  Perhaps we would find that the differences often cited as evidence often become indistinguishable.

Let’s move away from the debate and politics of education and towards a classroom lesson.  Consider major league baseball home runs.  The dot plot below shows the mean number of home runs by players on National League team rosters at the end of last year, made using Tinkerplots, the middle-school cousin of Fathom.

Mean HR's

The dot at the far right represents the Milwaukee Brewers, with the LA Dodgers just behind at 11.2  What conclusions can be made about the Brewers, based on this data point?  How many of the conclusions below are appropriate, based on the means?

  • The Brewers are the best home-run hitting team in the national league.
  • The Brewers have all of the best home run hitters.
  • The Brewers score the most runs.
  • The Brewers are the best team in baseball.

What do we know about the individual home run hitters on the Brewers, or any other team?  Not much.  If we plotted all of their individual home runs, what should we expect to see?  Will half be above 12, and half below 12….or can something else occur?  The plot below shows all hitters’ home runs from last year, with the Dodgers and Braves pulled out to demonstrate individual team distributions.Individuals

The Dodgers’ team average of 11.2 gives us a nice summary of the team performance, but how well does it describe individual performance?  Only 3 of the Dodger players hit near 11 home runs, while the others are distributed away from that number.  We also gain an appreciation for the right-skewness of the distribution, which the means did not reveal.  This provides a vital lesson that while the mean provides one overall summary of a distribution, it is not the only summary we should consider, and the mean tells us little about individuals.

Finally, we may also become impressed by high means in our first plot, as the scale of that plot shows clear “leaders”.  But what happens when we use similar scales for both means and individuals?

Similar Scales

Are we as impressed with our means leaders as we once were?  And while there are certainly teams that are ahead, but how much are they ahead?  What are the similarities and differences between the distributions of means and individuals?

I hope this demonstrates the rich discussions you can have with your students regarding the application of statistics.  Get your students writing about statistics, sharing conclusions, and discussing ideas.  The statistical future of our CNN anchors, government officials, and mothers depend upon it!

Categories
Statistics Technology

Let’s Play Plinko! At the Siemens STEM Academy

Today, it’s my turn to present at the Siemens STEM Academy in Silver Spring, after an exciting adventure in Washington where the fellows spent the day exploring the White House and the National Museum of Natural History, with a few of us escaping to the Air and Space Museum.

Air and Space

For today’s presentation, I took an old favorite from probability class, Plinko, and shared methods for using tech resources to share data and facilitate discussions.

To start, here is a video of Plinko drops I created, using clips from multiple You-Tube clips of Plinko games.

Before the simulation, I ask students to contribute their thoughts about the Plinko game, and the chances of winning the $10,000.  In the past, hand-raising or dry-erase boards were used, but the site Poll Everywhere provides a chance for feedback via cell phone text or website:

Poll 1

poll 2

I have walked students through Plinko simulations in my probability classes for over 10 years.  Students use flipped coins to travel down the pegged path, using a heads to move right and tails to move left. The slot the Plinko chip lands in is then recorded, and the experiment repeated.  Lately, I have encouraged participants to explore random.org, which provide random number draws, flipped coins, and cards drawn from decks.

Pairs and small groups generate data on Plinko success, which we would like to compile into one large class set.  In my earlier years, students kept track of their data, then contributed to the class results on my overhead projector. Later, the TI Navigator system was used to transmit results through graphing calculators. In today’s presentation, a Goggle form was shared for participants to submit results, then a summary table created.

Results

For submitting results, fellows were challenged to make a 30-second video which summarized their findings. Video were then submitted to my YouTube channel, using cell phones.  Look under your channel settings to find your submission address.  Here, Alysha and Brandon share their discoveries:

Finally, a photopeach slide show demonstrates how to get your class focused the next day, by summarizing an earlier day’s events through a photo slide-show quiz.

Using technology efficiently to keep students engaged, share data, and encourage innovation have been strong themes this week.  It’s never a bad time to examine lessons we have used for years and evaluate new technologies to improve their delivery.

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?