On the Difference Between Answers and Understanding

Both my Father and my Father-in-Law (z"l) were science teachers. Between my Dad's teaching non-majors and my FiL teaching high school, I'm certain they both strived to impart a love and appreciation of science to a skeptical audience. In their own ways, they imparted to me the fundamental difference between understanding versus simply having the right answer. My father-in-law would pour over physics textbooks the same way most people would re-read a favorite novel; all along the way relishing the fundamental. My Dad would pass along the warning he gave his students about spreadsheets: they'll always give you an answer, but is it the right answer?

Yesterday, as I fiddled around with the math behind stadiametric range finding I had a moment where this notion of understanding a topic crystalized. Typically, range finding is used to find an unknown distance. The forumula I was working with was as follows:

ad   od
-- = --
ah   oh

In every YouTube video I saw on the topic, the goal was to calculate the distance to some target, ad (actual-distance). But, it ocurred to me that you could use this same forumula and technique, yet flip around what you're solving for.

Say you were with a buddy and you wanted to run some 100 meter sprints. You could plug the known values into the formula above:

100m    66cm
----  = ----
5'6"     oh

In this case, we can solve for oh (observed-height). Doing this, comes to:

(100m / 5'6")        = (66cm / oh)  ; plug in the numbers
(10,000cm / 167.64cm) = (66cm / oh)  ; convert to cm
(66cm x 167.64cm) / 10,000cm = oh ; solve for oh
1.1cm = oh

This is what 1.1cm looks like marked out on my wallet-sized ruler:

I can now find 100 meters by walking away from my buddy and every so often holding this marked-off ruler out in front of me. When my running partner completely fills the marked off bounds of the ruler, I I've hit 100 meters. Science for the win!

While this may be a very minor bit of mathmetics, it made me smile when it all clicked for me. I'm confident both Dads would appreciate my little discovery and that I'm grasping their wisdom.

Over the years, I've watched as answers have become easier and easier to come by. Calculators, spreadsheets, Google, Wikipedia and Stack Overflow; oh how wonderful these services can be and how quickly they can reveal an answer. And now we seem to have reached peek-answer availability with Chat GPT. You can not only have the answer you want, but in your preferred voice and tone. Who could ask for more? And yet, understanding is still the gold the standard. Perhaps Shel Silverstein put it best:

The Homework Machine,
Oh, the Homework Machine,
Most perfect
contraption that's ever been seen.
Just put in your homework, then drop in a dime,
Snap on the switch, and in ten seconds' time,
Your homework comes out, quick and clean as can be.
Here it is— 'nine plus four?' and the answer is 'three.'
Three?
Oh me . . .
I guess it's not as perfect
As I thought it would be.