While celebrating my birthday last week at my daughter’s apartment, right between the birthday cake and the short walk over to Starbuck’s for my free birthday reward drink, something profound occurred that shifted my long-held perspective on vertical scales.

Those of you who follow my writing likely are familiar with my feelings on vertical scales. Perhaps the nicest thing that I’ve ever said about them was during a 2021 presentation: “vertical scales are nothing more than a neat technical exercise – a psychometric party trick, if you will.”

In one of the very first posts to this blog, I questioned whether vertical scales met the first requirement for reporting scales: to make test scores more interpretable and useful for the intended audience. In that post, I also suggested that vertical scales add more confusion than clarity to two of the most important asks that people are making of test scores; that is, providing information to support content-based interpretations of student performance and inferences about student growth.

I still stand by those statements.

So, in what way has my perspective changed?

What happened on my birthday?

Happiness Is A Warm Puppy

After lunch and birthday cakes, one small cake made by my wife and one by my daughter, it was time to open presents, which by the way included sweet hot and mild Capital City Mambo Sauce… iykyk

I began by opening the cards, as normal people do. The card from my daughter brought a smile to my face – a 3-D representation of the entire Peanuts gang with Snoopy smack dab in the center, as he should be.

After opening presents, I began to pack things up for the trip home. While doing so, I happened to look at the card from the side rather than straight on. That 90° shift in perspective changed everything. I was no longer looking at a smiling Snoopy.

I wasn’t sure what I was looking at. I start feeling a little dizzy and then nauseated. Frankly, the only word to describe the image before me was grotesque.

So, my thoughts immediately turned to vertical scales!

No, of course not. That would have been cool, but that’s not how the brain works.

How The Brain Works
(at least mine)

After a bit of processing, I realized that they had created the 3-dimensional effect by stacking the same image multiple times with a small gap in between each (picture a typical interim assessment administration cycle). What I was seeing from the side view were three images of Snoopy’s smiling teeth with the rest of him receding into the background.

Those image of Snoopy and the associated explanation made its way into my brain, joining fresh memories from NCME in Philadelphia: our Saturday session on years of learning and the related Sunday session on vertical scales; the long escalator ride up to the Terrace Ballroom, and the apple fritter from Beiler’s. The apple fritter doesn’t figure into the rest of the story, but it’s really good (blog sponsors gladly accepted).

As my wife, daughter, and I walked to/from Starbucks, Snoopy (scary), the escalator (scarier), and vertical scales (scariest) were churning in that virtual vast empty space my brain utilizes (picture the Terrace Ballroom during our NCME session).

The Philly escalator morphed into one of those escalators in the DC Metro – most likely Woodley Park going back to my first AERA conference in 1987. The DC Metro escalator morphed into the Porter Square escalator in Cambridge, MA – also long and on the Red Line. That led me to the set of parking garage escalators at Alewife, the final stop on that Red Line.

Exiting Alewife, you choose an escalator based on the floor where your car is parked. Starting at the same spot on the Concourse, each escalator goes to a specific floor. The escalators are stacked next to each other.

Then everything fell into place.

Escalators

Snoopy

Vertical Scales

Eureka! 

Cowabunga!

Can You See It?

It all makes sense now. Like the Alewife escalators, vertical scales are not one scale. They never were. They are a set of scales stacked next to each other, like Snoopy on the card.

Vertical scales share one common dimension (or axis) – the numbers that we assign to them.

But they are offset on a second dimension – grade level and everything associated with it.

And in the real world, there are more than two dimensions.

In a 3-dimensional real world, imagine my friend and I getting off the train and heading toward our cars – hers on the second floor because she left her house on time and mine on the fourth floor.

As we step onto the escalators at the same time, we will be at the same height, but separated by the distance between our escalators.

If the two escalators have the same pitch (rise over run, angle of inclination, etc.) and speed, we will remain at the same height until her escalator reaches the second floor while I continue on to the fourth.

Viewed from a different perspective, we may appear to be much closer together than we actually are.

Viewed from a certain perspective, we may appear to share the same space. (like when the moon blocked the sun a few weeks ago).

But we were never actually in the same place. We were always two people in two different places. We differed on another dimension – an important dimension.

It’s the same with vertical scales.

Vertical scales were never the same scale.

They start out as individual grade level scales and they remain individual grade level scales.

We employ our neat party trick to take points on those separate scales and assign them the same number on a certain dimension. Student performance, however, still differs on perhaps the most important dimension: content; that is, the competencies or standards that those students have actually been exposed to and mastered.

We have never really defined what that mystery dimension represented by the vertical scale is. Magicians and psychometricians never reveal their secrets.

Vertical scales seem plausible because our brain tells us we’ve seen this “2 into 1” maneuver work in other areas. We’ve seen rulers with inches on one side and centimeters on the other. An old thermometer might have F° on the left of the mercury tube and C° on the right. On a digital thermometer, we just tap a button to change scales. Right now in our cabinet we can pull out a measuring cup with ounces and cups on one side and milliliters on the other.

The little twist is that in all of those other cases, we are taking one quantity, entity, or “thing” (for lack of a better term) and are describing it with two different scales. A rose by any other name…

In the case of vertical scales, we are taking two different things and trying to describe them with one scale. Sleight of hand.

The length/height of something is the same whether we measure it in inches or centimeters.

The temperature is the same whether we express it in degrees Fahrenheit or Celsius.

Not so for student achievement.

The achievement of a fourth-grade student will always be different when measured with a fourth-grade mathematics test and an eighth grade mathematics test.

The achievement of one student measured on a fourth grade mathematics test will always be different that of another student measured on a fifth, sixth, or eighth grade mathematics test.

As for the measuring cup, we really should be weighing our ingredients anyway.

My best guess is that the mystery dimension on vertical scales is proficiency – as distinct from achievement – but let’s save that for another post.

For today, I’m satisfied with having arrived at a new perspective on vertical scales. What a nice birthday present.

I wish that I could celebrate with an apple fritter from Beiler’s, but I’ll settle for my free bakery treat from Panera. Can’t wait to see what inspiration I’ll get from a cinnamon roll.

Header image by StockSnap from Pixabay