#unitrebel Why unit analysis is overrated and why cancelling units borders on criminal

The following is my opinion based on 20 years of teaching physics at various high school levels from AP C to a trig-based to an algebra-based to a collaboratively taught special education version.  

A couple of weeks ago, I was participating in a white board circle during a modeling workshop.  The white board at which we were looking contained a notation that showed something that like: (m)(kg)/(kg) = (m).
In 'student-mode' I was like, "what happened to the kgs?"
The answer came as, "then cancelled."
I then asked, "what do you mean 'they cancelled'?"
What followed was a 20 minute session of 4 or 5 know-it-alls trying to convince me that kg/kg = 1.
Now, to be honest, I was being a little bit obstinate.  But my ulterior motive was to ask the question to the group:

"Why do we spend so much time talking about units?"
"What does it even mean to 'cancel' units?
"Why do we let our students cancel units?"

In my experience working with students of all levels, units mean very little to them.  That is a hard truth given how much they mean to us -  trained physics teachers and physicists. However, units are for people who already understand units.  When you watch an expert problem solver read and interpret a physics problem, select an appropriate equation, plug in the numbers with the correct units, cancel and solve, it looks like a maestro conducting a symphony.  However, is that our students?

Introductory physics students struggle at each step of this process.  And to ask them / require them to include the units in the problem solving steps and it really does just get in the way.  We would love if it promoted better problem solving or if it helped them make more meaning of a problem.  That would be AWESOME!  And I bet for some students it probably does.  But the students who get the units are already good problem solvers and don't need the benefits we would like the units to provide?

So what do I do about things like Newtons, Joules and Amperes?
When a new topic get introduced, usually through some sort of physical experience, the question of measurement and units comes up as a question posed by the students.  With force it is easy because the spring scales they use are calibrated in Newtons.  I would rather a student know how 1 N worth of force feels like in his hand than what the base units are.

Joules are very tricky.  In fact, there not a great way for students go discover these units.  So guess what?  I don't really use them!  The students decide that the measurable energy is calculated as the area between the graph line and the horizontal axis on a Force vs. Displacement graph.  So the units of energy, as described by the students are Nm.  Does calling this a Joule give the students more of an understanding?  I don't really know.
Current is more like force in that the multi-meter they are using has units of A already so we just go with it.

If, then sometime down the road, the students want to know something like, "what exactly is a Newton/Joule/Ampere anyway?"  Then we can get down and dirty because they (hopefully) have the contextual knowledge and practice to answer the question for themselves.

We know that real learning comes through discovery and conversation.  So any discussion of units has to have those two components.  If I am just the disseminator of information about units the students aren't going to get much out of them anyway.  So by providing less information and more opportunities for the students to experience unit analysis in context I feel like I'm doing my best for them.

So what about cancelling units?

My goal is always deep conceptual understanding.  There is some problem solving in my classes but much less than in traditional physics classes.  But there is some.  So how do I answer the question, "How are students going to know what the correct units are for the answer if they don't keep track and cancel along the way?"  I get this a lot when I'm up on my unit soapbox.  Let's first address one of the central issues:  why exactly to we think (and let students think) that kg/kg = 1?

I have no evidence or experience with this.  To be honest, I am not even sure that 4/4 = 1 except that I was told that in 4th grade and I've used it ever since.  I remember in every chemistry class I've ever had using dimensional analysis to solve mol-type-problems.  And although I was proficient I never really understood how or why it worked.  But it did and that was all that mattered.

Now, I know that right about now you're probably either yelling at the screen or thinking I'm an idiot.  But let's assume that I'm not.  How does 4/4 = 1?  Or even better how does 4 apples / 4 apples = 1.

People from here have told me stuff like, "well if you take for apples and divide them into groups of 4 apples you have 1 group!"  (with indignation).  But my question is, "where did the groups come from?  There are no groups in the original statement! And where did the apples go?  Are they still there?  Are they hiding?"

Have we, as a collective group of science teachers, allowed this process to continue without questioning its understand-ability?  And think, if I am having an issue with this, what must it be like for students?
Actually I know what it is like for students: mostly they don't care.  They like to cancel units because it makes things easy for them and they don't give a shit if they understand the why.

I want to know about the why.  And I want my students to want to know about the why.

If we now take the two concepts mentioned above; unit inclusion and unit cancellation and put them together we get to the problems I've encountered with students.

Let's consider the idea of slope.  In the classroom students measure the position of constant speed buggy cars with respect to a running clock.  If the car starts some distance in front of the zero line a graph can be generated by students with a slope and an intercept.  The equation may look like:

x = 35t + 20                                         or it can look like
x(cm) = 35(cm/s) + 20(cm)

Which of these is easier to understand?  I would assert that the top is easier to understand for students.  You and I will see the bottom and understand all of the syntax.  But a student would look at the bottom and ask themselves, "Why are there so many letters?"

If we then want our students to use these equations by plugging and chugging things can go from bad to worse.  I've heard lots of students look at the bottom equation and ask, "how do I know what to plug in where?"

If you study Newton's Law of Universal Gravitation you will invariably have to deal with the Universal gravitational constant.  The value is something in the range of 6.67 x 10E-11.  But the units are N(m)(m)/(kg)(kg)  When the students get to this value and try to use it to solve a new problem, they are very confused because of all of the letters!

Why are we, as a group, so set on keeping the units intact when all we've seen is confusion from many of our students?  Unit analysis does not provide insight into the underlying physics for introductory physics students.  I think it would be amazing if it did but alas it does not (for the majority of students).

In conclusion, the traditional methods with units are fine for some students.  But I want to know, is there a better way?  I hope there is.  And I'd love to let go of the trappings of traditional instruction and brave the new world.  Let's try and see what we can do.  Students today are different from those just a couple of years ago.  I want to change and adapt.  Let's not keep doing it the old way without a real understanding of the why.  And let's help our students develop methods that work for them from their own understanding.