Monday, May 22, 2017

In Orbit

     A number, I'm told, is like a Russian doll.

  This is because the quantity that a number represents contains the quantities represented by all preceding numbers.  This is called hierarchical inclusion, and its understanding is a very important stage in the number sense trajectory.  I thought I knew this.  But something happened recently that caused me to wonder: Do I really understand?
   It started in fifth grade with a Fraction Talk:

 Two different attempts caught my eye, and I turned them into a notice and wonder activity:

     I was trying to draw out the idea that the first response, the one on the left, was correct because the student had accurately labeled each small square as one-sixteenth of the whole.  However in the second response, the small squares were labeled incorrectly.  That student was counting by sixteenths, labeling each successive square as if it included within it all the preceding squares.  Was that an example of hierarchical inclusion?  I went back to this picture, which has helped me understand the concept:

From Early Childhood Mathematics Education Research: Learning Trajectories for Young Children by Julie Sarama and Douglas H. Clements
     It certainly seemed the case, but then I noticed something in the caption that I hadn't really noticed before; that each cardinal number includes those that came before.  What about other kinds of numbers?  Wait a minute.  There are other kinds of numbers?
    A quick search led me here, where I learned about three different kinds:

  • Cardinal numbers.  They tell us how many of something there are. 

  • Ordinal numbers.  They tell us the position of something.
  • Nominal numbers.  They are used as names, or to identify something.

     I realized that I already knew about ordinal numbers; nominal numbers were new to me.  So I started collecting numbers around my school and mentally trying to classify them.  Here's a sample of what I found:


  Nicole has her first graders count the days in school, and my first instinct is to call this a cardinal number.  Many primary teachers accumulate tokens, such as Popsicle sticks or unifix cubes to represent each successive day, suggesting that days are a set of things that can be counted.  However does each individual day include within it all preceding days?  Or do we get a fresh start each morning?  I don't know.  Here's Sarama and Clements, again in Early Childhood Mathematics Education Research, "When the topic of 'ordinality' is discussed, even by some researchers, it is often assumed that all ordinal notations must involve the terms 'first, second, third...' and so forth.  This is a limited view.   A person who is 'number 5' in a line is labeled by a word that is no less ordinal in its meaning because it is not expressed as 'fifth.' (p. 85)   So here Day 124 is the 124th day of school.  It is in position 124 of a sequential count of the days of school that started at 1 and will end at 180.
Conclusion: Ordinal.  Maybe cardinal.  Definitely not nominal.


   This storage room doesn't include within it all rooms numbered 1 through 139, and there's no reason why this specific room should be identified as the 139th in a sequence of rooms.  In fact, there are not even close to 139 rooms in my school.  139 here seems to function as a signifier or name for this particular room.
Conclusion: Nominal.


    The hooks in Wendy's grade 3 classroom are in sequential order.   The polka-dot backpack is on hook number 3, or the third hook.  In one way it seems like the number 3 is also acting here as a name for that hook.  Can a number be both ordinal and nominal at the same time?
Conclusion: Ordinal.  Maybe nominal.  Definitely not cardinal.


     The thermostat in the office must be broken.  No way it was anywhere near 76 degrees.  Regardless, this number is definitely not nominal.  Thinking of the temperature as I would see it on a mercury thermometer...

...helped me see that the number of degrees did include within it all preceding temperatures.  So did this, which I recorded one night testing my chicken pot pie:

thermometer from Joe Schwartz on Vimeo.

Conclusion: Cardinal.


 The back of a fourth grader's basketball jersey.  We're not counting or ordering anything here.  The number 3 is just identifying Clippers point guard Chris Paul.  Maybe Paul picked it because it was his favorite number.  Maybe he wanted another number but it was already taken.  Anyway, the student sporting his jersey isn't even a Clippers fan.  He likes the Cavs.  Front runner!
Conclusion: Nominal.


   This is how the kindergartners in Kelly's room keep track of who's in school.  This one was easy.
Conclusion: Cardinal.


     10:36:16.  What kind of number is this?  I've come to think of nominal numbers as having a randomness about them; a phone number, a driver's license, an account number, my zip code.  I don't get that feeling here.  Could this be ordinal?  Ordinals have a sequential, positional feel to them, and of course there's always the th.  Could we say it's the 16th second of the 36th minute of the 10th hour?  Is 10:36:16 a Russian doll, nesting right between 10:36:15 and 10:36:17?  Maybe it's like what T.S. Eliot wrote:

    Time present and time past

Are both perhaps present in time future

And time future contained in time past.

Conclusion: Cardinal.  (Pretty sure.)

     Sarama and Clements cite research indicating that, "It is not until age nine that most master the hierarchical inclusion relationship."  (ECMER, p. 339)   If I did master the relationship way back when, then I guess my path through the world of number sense is more like an orbit than a trajectory.  It's taken me 46 years, but I've circled back around to find that numbers, for me, remain enigmatic and mysterious, in need of continued and constant rediscovery.



  1. You brought back one of my worst memories. In atomic orbitals, electrons filled them in an order of "s, p, d, ..." To me those letters were numbers. (Think Hebrew.) They were nominal with ordinal aspects. When I mentioned this to the professor and several grad assistants, they snickered at me and said I was misunderstanding: "Letters aren't numbers. Are you nuts?" I persisted because I was stunned people not only didn't know cardinal vs ordinal vs nominal, but directed hatred to the very "waste of time" in mentioning such a worthless concern. I was crushed because I thought I find at least one "smart" person who would agree, even if bored. Not one, and this was Stanford. A low point for me.

    1. Sorry to bring back this painful memory, Dennis, but thanks for sharing the experience. I like your description, "nominal with ordinal aspects." That's how I thought of the coat hook numbers. Maybe the Stanford folks could have used a little course on Gematria.

  2. does each individual day include within it all preceding days?

    Like the rest of this post, that line is pure poetry. I love this, Joe.

    The clock question is the one I keep thinking about. The thing that's bugging me is that clock numbers are like modular numbers. Can a modular number ever contain everything that comes before it? What does come before a modular number?

    If 11 + 1 = 0 in modular numbers, then does 0 include 11? No way, right?

    1. Thanks Michael! I'm not even sure what modular numbers are! But if they reset every time, then no way would 0 include 11. Time might function differently? Would counting the days in school be like that, too? 0 to 180 and then back to 0 again?

    2. The clock example had my wheels spinning too! As I think about it's classification, it feels like it could be cardinal or ordinal depending on the context. If I were considering "how many hours it has been since _____" or "in how many hours it will be _____", that feels cardinal. But when I think about the 11th hour, the context might lean more toward ordinality. I am now thinking that all the numbers we see must be considered within context to classify them. Thoughts?

    3. I agree. I think that context has to be taken into consideration. Just a naked number can't be classified without context, though I think most times a naked number is taken to be cardinal.

    4. So it's not numbers but uses of numbers that can be classified?

    5. I didn't think of it that way but yes, that's a useful way for me to think about it!

  3. I love this post Joe. I feel like you're digging into the meaning of these kinds of numbering - on the sort of level I, and, I guess, all of us need. Keeping it simple. What's an example of...

    So ordinal is like things in a line. Cardinal is counting things and saying the number is the last number you count, so that all the other things are inside like Russian dolls...

    Then you turn to the thermometer. The numbers are on a line, but do the temperatures include the lower temperatures? Does a comfortable 26°C day somehow *include* a chilly 3°C day? I guess if we have a physicist's view of temperature, we could say that the temperature represents movement; there's already some movement, then we add more movement on...

    I'd like to get really clear about this. I hope someone really expert comes along and explains it all in really simple terms for us!

    1. Thanks Simon! Yes, what I thought would be a pretty simple exercise turned out to be anything but. One thing I thought about but didn't explore is whether or not we could make any generalizations about the kinds of numbers used to measure things. For example, if you draw a line 7 inches long, you need all preceding inches, and it certainly includes 1, 2, 3, 4, 5, and 6 inches too. A box that holds 10 cubic inches holds all the preceding cubic inches as well. So those would be cardinal. Temperature is measuring, and so is time, so maybe they're cardinal too?
      Agree we need an expert, which is why I'm hoping that Douglas Clements will chime in!

  4. I love this post! I wrote a post in the beginning of the year about how I wonder if third graders re-visit the counting and cardinality standards when they begin to study multiplication ( Now, you've got me thinking about fractions. It is like every time the unit changes, we have to revisit what we know about how the units fit together and what the last unit tells us about the rest of the unit. I am still wondering about the sixteenths. What does the student who labeled the pieces as a count know about fractions? And the modular conversation is fascinating. I have to revisit modular numbers, but I am wondering about the "next unit". So after 12 hours, we go back to the first hour, but those 12 hours now live inside a different unit called a "day". I absolutely love this conversation. I am going to go on a number hunt all day today.

    1. "What does the student who labeled the pieces as a count know about fractions?"
      As I was writing the post I had a conversation with Graham Fletcher about this, and that was the exact same question he asked. He argued that you could label the pieces as a count and still understand that each piece was still only 1/16 of the whole. But I'm not even sure that the categories apply to fractions, although it seems they they must.

  5. This was a great read and my thoughts are spinning. I will likely be trying to classify numbers all day.

    I remember working with number line vs number ladder this year and trying to classify which one would represent ordinality and which one cardinality. A ladder consists of spaces, no zero, numbers are like counting blocks occupying space. A number line can be infinitely broken into numbers and makes me think of more of a number river than number path. It also has a zero. I thought of a number line as representation of cardinality, where numbers do contain within them the previous numbers. And ladder seems to represent number blocks, the counting step,ordinality. But then counting objects is actually ordinal relationship, not cardinal.
    We were thinking about the time with my students, is time and calendar more like spaces or lines? We thought about ordinality of our calendar without zero, with days and years like blocks. But now after reading this post and the comments I realize that the units for our time measurement are nested within each other. There is a modular feel to them, but the minutes don't just reset every hour; they switch gears for hours, hours push days to move, days reset to change months and 12 months start over when we add one to the year count. Like Sarah mentioned, it actually seems like a place value system. I wonder if the time itself has the features of cardinality but the way we capture it is an ordinal system?

    Time to go to work and to sort numbers. This conversation is fascinating, I'd love to continue it.

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