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Telling the full story of a number

September 12, 2011

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 6.02\times 10^{23} 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 \frac{1}{3}, 0.\bar{3} or 0.3333333333, 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 4\times 5=20 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.

11 Comments leave one →
  1. September 13, 2011 6:27 am

    John:

    Thanks for the post. Numbers do tell stories and sometimes we don’t spend enough time telling them, and the history surrounding the number. Students need to understand that they don’t materialize out of thin air, especially numbers like Avogadro’s number or Planck’s constant. Their is an elegance associated with many numbers, even those that are simple. By the way, Avogadro’s number is 6.02 x 10^23 atoms per mole. And then what is a 1 mole. Well, it is defined by Avogardo’s number, as well as the number of grams equivalent to the atomic or molecular weight. Thanks for the post and maybe correct your number so all the chemists of the world don’t inundate you.

    Thanks!

    Bob

    • September 13, 2011 6:54 am

      Thanks Bob,
      I should have caught that, but what’s a power of 10 between friends?🙂

  2. September 13, 2011 8:08 am

    I personally shy away from being the unit police, especially when student work represents them working something out for themselves (and I just happened to collect it). I want the issue come up in discussion or in writing when there’s a real audience (i.e. not teacher) who might be confused or might need to know. For example, last week students were working on my goal-less video problem with the speedometer; One groups got like 6,000 mph per hour and another group got like 1.67 mph / s and another group got like .0005 miles per second per second. We talked about how and why they got different numbers and if they meant the same thing or if they meant a different thing? It was a struggle to talk about what 6,000 mph per hour meant. A minute later, another group was asking for help, because they solved for the distance two ways and got different answers. Quickly another student realized that while they had converted the acceleration into miles/s/s, they had left their speeds in terms of mph. They spotted the unit problem before I did.

    • September 13, 2011 9:31 am

      I totally hear you. I don’t want to be the unit police, and I really don’t want to be the sig fig police (especially since sig figs is a word I never use). But I’ve found that the act of writing a number is something students to almost without thinking, that they need incentive to really stop and think about the number they are writing. So far, I’m sad to admit the only incentive I’ve found is the “you don’t have full understanding until you set up a habit of telling the story with every number” and the grade stick that implies.

      • September 13, 2011 9:52 am

        I suppose what I see is that we both want the need to both care and be careful with units to be authentic –not because “our teacher makes us” and not because “naked numbers are bad”. The first is problematic for me because of the authority-driven nature. The second one is problematic for me because its divorced from authentic epistemological concerns.

        • September 13, 2011 9:57 am

          I’m totally with you. I think I wrote the original post I wanted a way to get past both “teacher forces me” and “naked numbers” to an intrinsic motivation to tell the full story of a number. And I’ve had lots of moments like what you describe where students find the value of telling the full story of a number through a lab or discussion, and these are the moments I strive for, but so far, I find they are not enough to promote the day-to-day attention to this important detail.

  3. September 13, 2011 9:01 am

    when i taught physics, i could never seem to motivate units/sig figs really well, but this seems like a perfect way to frame the discussion and a perfect mantra for the students to keep in their head. every time they write down a number thinking that they are telling a story. thanks for sharing, i’ll apply a modified idea to word problems in calc this year

  4. September 13, 2011 4:32 pm

    Not quiiittteee what this topic was about, but if you missed it I wanted to share Geoff’s link here http://pedagoguepadawan.net/124/measurementuncertaintyactivities/ about establishing measurement uncertainty in his class. I dig this a lot. Instead of arbitrary lopping off of numbers, his students figure out what the best measurement they can reasonably get in that class with those tools.

    • September 17, 2011 12:08 pm

      Jason,
      Thanks for reminding of this awesome activity from Geoff. I’d seen it before, but forgotten about it.

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