An unnoticed rule for number language

I am coming to the end of my big place-value extravaganza in the math course I teach for future elementary and special education teachers. I had a conversation after class last week that reinforced the importance of the time I spend on place value.

Over the last two years, I have adapted a system for teaching place value in this course that comes from the ideas of JoAnn Cady and Theresa Hopkins at the University of Tennessee, Knoxville. The system, which they call “Orpda” and which I have bastardized as “Ordpa” (and two years in, it’s too late to change now!) is a base five place value system that depends on a new set of numerals-seemingly random symbols from the top row of the keyboard. But more than that, Cady and Hopkins developed an inquiry approach to using the system. As an example, the new symbols are introduced (in order in my version of the system: @, #, $, %), a symbol for 0 is introduced (!) and they ask students, “How should we represent the next number?”

From experience, very few people think of using place value. Many will invent a new symbol. Others will add the existing symbols, suggesting things like “#+$”. Still others will draw an analogy to tally marks: @@@@@. The place value answer to the question is extremely rare in my work with future teachers and in presentations I have done at state math conferences and professional development sessions. I have learned through this system that our minds are not programmed to think in terms of place value. Instead, our minds are programmed to think additively.

I have enjoyed this side of working with the Ordpa number system-it helps us to understand children’s challenges in learning and using the decimal number system. This was what Cady and Hopkins developed the system for.

But I have become even more interested in how the system highlights number language. I have written about this in other places, including a forthcoming article in for the learning of mathematics. But last week a student asked a question that I had not previously considered.

We have named our first two-digit number: @! as flop and we have named our first three-digit number as flip. Today, we were considering the difference between flip flop and flop flip as a way of understanding a video we have watched in which a young girl says “six hundred plus four hundred equals ten hundred” and then writes “110”.

My student asked after class, “So how can flip flop mean flip plus flop, but flop flip means flop times flip or flop groups of flip?”

What a wonderful insight! Consider the construction two-thousand two. We don’t think about the fact that the first two is multiplied by the thousand (“two thousands”), while the second two is added. And we certainly don’t think about the implicit rule that when a smaller number word precedes a larger one, we multiply and when the smaller number word comes second, we add.

Other examples that highlight this rule include 1100 and 100,000.

So (Wump) hats off to Cady and Hopkins for sending me down a path on which I continue to learn and which has clearly gotten my students thinking harder about elementary mathematics.

Incidentally, I’ll be presenting on the system, and on children’s and adults’ learning of place value at the National Council of Teachers of Mathematics Regional Conference in Minneapolis in November.