To examine a rule for number language that most adults don't think about, but which is essential for young children to master as they learn to count.
This is a high-level packet that assumes the reader is interested in detailed issues of place value and number learning. It is not intended to be a primer in any of the related issues. Links are provided for more information on all aspects of the text.
Over the last two years, I have adapted a system for teaching place value in a math course I teach for future elementary and special education teachers. The system 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 four 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.
In the course I teach, and in this packet, I use this new number system to closely examine the language of place value. How do we read numbers aloud?
Cady and Hopkins didn't just develop a goofy set of symbols, they 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 and national 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.
The place value answer is this; we will compose the symbols @ and ! to get @! The "@" symbol represent 1 group, the "!" symbol represent no ungrouped units. We read this number in my classes as "flop". The next few numbers in the counting sequence are, @@ (flop at), @# (flop pound), @$ (flop dollar), @% (flop percent) and #! (pound-flop).
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 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. On this day, 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 (one-thousand one-hundred) and 100,000 (one-hundred-thousand), or 205 (two-hundred five) and 502 (five-hundred two).
The Ordpa number system helps to remind us that number language is challenging to learn.
A student was struggling recently with the counting sequence #@ (pound-flop at), ## (pound-flop pound), etc.
"Why isn't it just 'pound-at'?" she asked.
"Why isn't 21 just 'two-one'?" I replied.
"Because of the pattern," she answered. "It's twenty, twenty one, twenty two, etc."
But it isn't just the pattern. The number language corresponds to place value. 21 is 20+1, so we say "twenty one" to indicate this, and furthermore "twenty" is "two tens".
It's exactly the same in Ordpa-it's not the pattern that matters, it's the place value structure of the number system. #@ is #!+@, so we read the two parts of the number, "pound-flop at".
If we don't examine our own number language, we cannot help students through their struggles. The Ordpa number system has been extremely helpful in getting future teachers to examine their number language. See the links above for more details on the system, including the original article in the journal Teaching Children Mathematics.