Telling the full story of a number

I tell my students every number in science tells a story—the story of a measurement. The way you write the number tells the reader how it was measured, and the units you use tell the reader what it was that you measured.

I use a number of stories to get students to think about this. I suggest that they think of writing a number down with the same effort it takes to spell the number out in cursive on a check—One thousand, one hundred and fifteen (antiquated, I know). We talk about how arabic numerals really are a tremendous gift for allowing us to communicate very complex concepts atoms in a mole, in a tiny number of marks on the page, and to imagine how much harder it would be to express ideas like this using Roman Numerals, or even hash marks.

It is really hard for students to engrain this habit of fully telling the stories of the numbers they write. In the era of cheap calculation, it’s far too easy to write , or , even if the latter requires 10 times more ink than the most thoughtful answer. We sort of make a big deal out of it—we call numbers without units naked numbers, and I hide my eyes whenever I see them.

To further help them with the habit, I tell my students that on any concept involving a calculation, they haven’t shown full understanding until they solve the problem while telling the full story of every number, even those in intermediate steps. Even if a student writes off in the corner, I consider this less than full understanding if they don’t clothe their numbers with units. I do this, because I think it’s a lot easier to build the habit of “I will tell the story of every number I write” than it is to build the habit of “I will tell the story of the last number I write,” and when students get in the habit of writing the complete story of their numbers, they’ll see that it is one of the most powerful aids in problem solving they have.

Two problems have come up with this approach however. The first is that too many of my students see their problems as “writing naked numbers” rather than telling the full story of the number. This leads me to believe they aren’t getting the big picture.

The second problem with this is that we haven’t really gotten a great way to illustrate the value of precision in measurements the lab. Our work with buggies really doesn’t require great precision, and when they find that their prediction is off from the actual result, they chalk it up to “human error” (my pet peeve) or the fact that the buggies don’t move in straight lines. What I really want is an activity early on that pushes them to see just how precisely they can predict something. And I realized too late that we really have that with Dan Meyer’s boat in the river problem. If you don’t measure the time very precisely, by stopping the video and scrubbing to the moment when Dan reaches the top of the stairs, you’ll end up with a time that’s wildly off.

Still this isn’t quite what I want from the lab—I’d like for my students to see two things in working with measurement:

• Improving precision of measurements can allow us to discriminate previously unseen things. For instance, it was improvements in our ability to measure time using atomic clocks that gave us experimental proof of special relativity.
• Improved prevision can’t be had for free, it comes with costs, usually in terms of time and cost. Nobel prizes have often been won for adding a few decimal places on to measuring one quantity or another, with atomic clocks being just one example.

That’s were I am now. Even though I am telling my students to tell the full story of the numbers they write, I don’t think I’m giving them an opportunity to appreciate the full story for themselves, and so for now, these guidelines become simple must-do’s to avoid losing points on assessments, which is totally where I don’t want to be. So maybe some of my readers will have some insight on how to move this idea forward.