Power Law, Bell Curves and 100%

In 2014 all students could read and do math.  Fact.  That’s what the No Child Left Behind (NCLB) law promised.  100%!

No, in reality, we fell short.  We did as well as law enforcement solved the crime problem–our focus got better (standards!  Common Core!), we used new tools (statistics!) and the numbers went down.  It brought a lot of needed focus on those groups normally ignored, especially those from lower socioeconomic backgrounds.  I don’t know if we’ll ever fix either problem 100% because, humans.

The Bell Curve

Prior to NCLB the “bell curve” was the norm.  Here’s a picture of it.


That is, a few kids got the “A”, a few kids failed, and a whole bunch took up the middle.  I remember my sister, in her physics class, would purposely get a few problems wrong because she would otherwise mess up the curve.  I had the same teacher, and getting half correct was good for a “B”.

Those days are (mostly) gone.  With the introduction of the rubric educators, began to realize that the goal of education should be 100% of students meeting the standard.  Parsing out a few “A” grades does not show rigor, but control–a great teacher should get all students up to their high standard.  The focus became on finishing the marathon, not so much worrying about who finished first (or last).  The bell curve, as far as measuring achievement, has (mostly) been phased out.

In the early days of NCLB our school became really interested in the “stanine”.*  The stanine is where that bell curve is divided up into 9 sections, based on standard deviations.  The “norm” falls in sections 4-6  and accounts for about half the students.  You’ll note that a few kids are a 9 and a few kids are a 1.  We used it a lot and the stanine was in the data sheets we always got.

At some point, someone realized that while it told us where the student was in relation to his or her peers, it did not tell us much about where they were in relation to the content.  After all, “In the kingdom of the blind, the one eyed man is king!”

That was when the age of the rubric and standards arrived to save us all.

The 100% of the Rubric

You can’t have no children left behind AND use a bell curve.  The bell curve assumes some kids are at the bottom.

It is important to understand the philosophical shift from one to the other.  There are many.  With the bell curve you not only assume that there are kids who will fail but also that you can only have a few superstars.  A graph like this should report known information, but too many see it as destiny–only a few can be stars.

In fact, in the bell curve world, if there are too many stars it diminishes the value of that designation.  Our entire discussion on grade inflation is centered around this point of view–only a few should be getting “A” grades or grades have no value.

Use of the bell curve also slots kids.  I was a “B-” student.  Educators crammed kids into the curve instead of looking at the data they had.  The data should determine graph!

At some point someone asked “What is an ‘A’ anyway”?  Thus began the movement towards rubrics.**   What helps me explain it to parents is the metaphor I hinted at above–the marathon.  Only one person can win a marathon, but nearly anyone can run and finish one.  Our school system is built to get everyone running, so why are we still using a stopwatch?  The rubric is the race’s distance.

Philosophically, having a rubric means all students can meet the standard.  Or, to use the metaphor, finish the marathon.  Some, with IEPs, might need support or a lot more time, but all can do it.  A plethora of “A” grades means that everyone is meeting the standard (and that the bar needs to be raised a bit, but that’s another discussion).

Of course, having a rubric means you have to define the standard.  And what “success” looks like.  The days of “I’ll know it when I see it” grading have faded.  Students and parents now ask for the rubric.  Teachers offer “exemplars” of good work.  Schools began “calibrating” and double scoring to make sure a “3” essay was a “3” essay no matter who grades it.  Grading became fair, or at least more objective.

The philosophical shift continued as letter grades began to be replaced with the 1-4 score.  Letters are a leftover of the old bell curve, with a history that parents intuitively and reflexively know and react to.  In fact, our grade level kept letters for that reason–a parent saw a “C” and they’d act, lean on the kid and come in for a conference.  A “2” never has the same effect.

Those words changed, too.  A “2” is for “approaching” while a “1” indicates “starting”.  Compare that to the “F” for “fail”.  There is little growth mindset in an “F”.***  Thanks to Carol Dweck educators now see students as pliable.  We talk about growth and fixed mindsets, even if not everyone (students, parents, some teachers) got the memo.  Anyone can do it.  No child SHOULD be left behind.

The philosophical shift of rubrics and no child being left behind has bloomed into other ideas, most notable the Proficiency Based Learning (PBL) movement.  Some notable changes include moving from a 100 point scale to using 1, 2, 3, 4 with targets, removing averaging in the “0” for work not done, and the idea of practicing a skill with formative assessments before offering the summative.  These have spawned the “re-do” and no-homework movements.  Full disclosure, I agree.  I agree with whatever means kids leave my classroom with the knowledge.

Of course, upsetting the applecart of traditional “winners” and “losers” rankles a few (mostly, the traditional winners).  The bell curve is great for those on the right side of the curve.  In a fixed mindset world, even those throughout the bell curve accept their fate.  Besides grade inflation concerns, this whole movement has been lumped into the complaints that if “everybody gets a trophy” then they are worth little.  If you adhere to the original orthodoxy, that’s true.  At our 8th grade “graduation” we stopped giving specific academic awards.  Instead, we celebrated a high level of achievement with the “Presidential” award and then let each kid identify a way they excelled over the past year.  No one in the audience felt it cheap.

While the PBL movement has done a lot to push all kids to succeed using a growth mindset, how can schools look at data in a meaningful way?

Power Rule

Vilfredo Pareto was a 19th century Italian economist who first articulated the 80:20 rule.  He showed that 80% of the land in Italy was owned by 20% of Italians.  Joseph M. Juran later theorized that 80% of effects come from 20% of causes. and named it the Pareto Principle.

As I look at the data laid out before me, I wonder how often I see this.  80% of my time seems to be taken up by 20% of my students (this is true in both academics and behavior).  80% of my students seem to meet the standard while 20% struggle.  It certainly paints a nice picture.  My fear is that, like the bell curve, we begin to slot kids.  If 20% of the kids are destined to fail, I’m off the hook of expecting ALL kids to make it.  If we are speaking of equity, why do 20% of kids get my attention (if you have your own kid, you know you want your kid’s share of the teacher’s attention).  I try and remember: Fair does not always mean equal.  Still….

But the 80:20 rule is something to keep in mind.  For example, my wife covers the papers of her students in comments.  I make about 20% worth of comments that she does, and I suspect (anecdotal) that it gets 80% of the effect.  Perhaps, 80% of the learning is due to 20% of my teaching (probably the lesson on Tuesday).  But, just as there are 20% that might not be meeting the standard, it might also show 20% are above the standard and need more.

Ah, I’ve just make a defacto bell curve.  That is the danger here.  Once the 80:20 rule starts become a rule more than a way to state what you observe there is a danger of fixed destiny.  As Sara Connor says, “No fate.”

So how about “power law”?  Power law demonstrates a relationship between two quantities.  The change in one results in a change in the other, but as a power of the other.300px-long_tail-svg  Like most things, it is best explained by Wikipedia.  To use their example, increase the side of a square and the area increases squared.

Here is a nice graph of it, and the 80:20 rule, too (green:yellow).

If you think in terms of power law, you begin to think of two things.  First, what variable can you change to get exponential change on the other end.  Not assigning homework, for example, and having kids write in class made a huge difference in learning for my students.  Second, power law makes one look at the tail–that’s what the 20% is called (in the graph, you can see it to the right, and it’s not always 20% or even close to that).

That 20% will take 80% more effort to move.  But, you could find the 20% change that moves it 80% results-wise.  Of course, I’m having a bit of fun here–if only it were that easy!  But playing with those numbers breaks the rut we educators put ourselves in.

Let’s look at using this graph in a very different way, though.  What if the 80% was a student being successful and 20% falling short.  How to measure that?  Traditionally, that would be an average.  But averages are fickle.  We distinguish between the median and mean in case Bill Gates walks in and someone thinks I’m a billionaire because that would be our average salary.  In our PBL world we are moving away from the average.

Does that far right end of the tail show the story?  Of course, in the grade book, that extreme tail score would be in there.  It might be a lesson learned.  But the story of competence is the 80% as much as the 20%

Our current grading program, Jumprope, allows for power law to be used in grading.  In short, the weight of older grades shrinks.  The more recent grades weigh the most, because we care more for what students can do today.  I thought it was the best between a history of competence (average) and what they can do now.

Except, if the last attempt was a bomb.  Thus, a history of success gets marred by a stumble.  Fair?  No, but such snapshots rarely are.

Where Does That Put Us?

Changing.  Always.  That’s where it puts us.  Education is a forever war.  And why is that important?  Because anytime an administrator or consultant (or both!) comes at you with charts, be skeptical.  I love graphs and data and especially predicting (which is a lot of folly, but fun folly).

But I got human being in front of me.  That’s who I gotta teach.  And believing in a growth mindset, each is a forever battle in our long, wonderful forever war.

Suit up.

*Some called it the “Stat 9”.  The story I heard was that, during World War II, the Army was looking for pilots.  They wanted to rank them on a ten scale, but with the limited, new IBM computers using the tens column would require a lot of extra power–so they went 1-9.

**In our Proficiency Based Learning (PBL) paradigm our school is now using the term “target” instead of rubric.  The target is the next step towards proficiency, while the rubric is a static tool in a document that, with a little tweaking, becomes a target on professional development days.

***Our school’s solution to the “F = fail” issues was to use an “N” for “no evidence”.  Parents always asked, “What the heck is an ‘N’?”  

Not knowing an “F” was for “fail” students often asked why it went from “D” to “F”–what about the “E”?  At one school I worked, they had an “E” which was “failing with effort.”  Instead of blowing work off, their work was marginal.  Because of how the final yearly grade was calculated, an “E” for the semester averaged in better than an “F” and allowed hapless kids to get credit if they rebounded with a “D” or better in the other semester and final exam.


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