What's a "second squared," anyway?

Read any textbook*, and you'll see the units of acceleration written - correctly, if suboptimally - as m/s².  

*Okay, any *physics* textbook.  Probably "A Short History of the United States" doesn't write the units of acceleration this way.

Most practicing physicists understand that acceleration is defined as the change in velocity per unit time.* That makes the acceleration unit (m/s)/s.  Of course a practicing physicist, someone who is quite fluent in mathematics, resolves the improper fraction by turning division into multiplication-by-the-reciprocal: m/s x 1/s = m/s².

*By the way, new physics students are just as confused by the phrase "per unit time" as "per second squared."  Just say "per second" or "in one second."

But our students are certainly not fluent in mathematics, nor in the physical meaning of acceleration.  Years ago, I asked advanced upperclass students this multiple choice question, in which the distractors were taken verbatim from student responses:

Which of the following describes the meaning of an acceleration of 1.60 m/s²?

(A) The elevator gains or loses 1.6 meters per second of speed each second

(B)  The elevator gains or loses 1.6 meters each second

(C)  The elevator travels 1.6 more or fewer meters each second

(D) The elevator travels 1.6 m/s² more or less each second

(E)  The elevator is either speeding up or slowing down by 1.6 meters for every second squared.

Our students, our naïve padawans, hear "meters per second squared" and assume that a "second squared" must be a thing.  The fact that most of the class got this quiz question wrong showed me conclusive evidence that the typically-written units of acceleration are an unrepaired pothole on the road to progress.

What does acceleration really mean, anyway?  In language my students understand: acceleration tells how much an object's speed changes in one second.  

That's often written as a = Δv/Δt, though this equation further obscures meaning.  Our students aren't fluent in math!  Either they don't understand what the delta means, and read the equation as "acceleration is velocity over time"; or even if they do understand that the delta means "change in," they have to laboriously translate that language in their minds before comprehension sets in.  Watching students process the mathematics reminds me of my 1992 Russian language class, where I earned the worst scores in the class on oral exams.  I had to translate what I heard into English (sometimes using mnemonics or slow recall to attempt to figure out the meaning of newer vocabulary); formulate a response in English; and then translate back to Russian.  This is not a recipe for scintillating conversation, any more is it a recipe for expressing physics knowledge.

Use m/s/s as acceleration units.  This is a completely correct expression.  Every time a student writes the unit and mentally pronounces the abbreviation - "meters per second per second" rather than "em slash ess slash ess" - that student is reinforcing the physical meaning of acceleration until it becomes natural.

I've been using this expression for acceleration units for nearly a decade now.  Of course it's not foolproof!  There is no One Weird Trick for teaching the most difficult concept in all of first-year physics.  Yet, I get way, way better understanding now than I did in the days when I said "meters per second squared."  Try it - you will, too.