Or

Author:
Christopher Danielson

To introduce the basic structure of a new place value system that was designed for the mathematical development of future teachers.

This packet supports student activities in a math course for future teachers.

Before reading this packet, consider spending a few minutes with my packet on composed units.

Tutorial

The Ordpa number system is adapted from work by Joann Cady and Teresa Hopkins at the University of Tennessee. In my courses for future teachers, we study this number system for two reasons:

(1) To learn about the structure of the usual base-10 (decimal) number system, and

(2) To better understand how people (especially young children) learn the decimal number system.

I do not claim that the Ordpa number system is a better, more logical or more efficient tool for counting or operating on numbers. Instead, the system helps us step outside of what we know to examine the mathematical structure of number, and to understand the learning process.

This packet presents the basic structure of the Ordpa number system. There are six sections:

- The first numbers
- Practice
- A hard question
- Place value
- Counting to larger numbers
- Next steps

Below is a chart shows the symbols we use for the first few counting numbers in the Ordpa number system. On the right is a number of objects, on the left is the corresponding digit. So in a sense, "@" means "1". But converting between Ordpa and our usual number system will not be helpful for our purposes, so try not to mentally convert these digits. Instead, learn to associate each digit with the correct number of objects.

This memorization process will be greatly helped by having a way to say the symbols aloud. Just as we learn the word sequence, "one, two, three, etc." We need to learn the number poem for the Ordpa number system. The next table includes number language-the word for each symbol.

Before proceeding, practice counting, "at, pound, dollar, percent...at, pound, dollar, percent..." Repetition is key to memorizing arbitrary lists.

Notice that the number names in Ordpa are better aligned with the symbols than they are in English. You already know the "@" symbol as "at" before reading this. Young children learn the number word first-say "three" through their experiences with language. But learning to identify that word with a quantity, and learning to associate it with a symbol require new learning. So in that sense you have it easier learning Ordpa than children do when learning the decimal system.

Now practice...write the correct Ordpa digit representing the number of objects in each cell in the table below (the cell in the upper left has @, the next one to the right has $, etc.)

And check your answers below.

So now you can count to percent (%). But how should we represent the next number? How should we write this many objects?

There are a few standard categories of answer: (1) Make a new symbol, (2) Add together old symbols, and (3) Juxtapose old symbols. Let's consider these individually.

(1) **Make a new symbol**. While a simple solution to the problem at hand, it's not really useful in the long run. If we have to make up a new symbol every time we add one more object, we'll be in trouble very soon. How will we remember the difference between the symbol for fifteen things and the one for sixteen things? The load on human memory is too great for this to be a useful strategy for developing a number system.

(2) **Add together old symbols**. This is a bit better, but things get messy quickly for two reasons: (i) Large numbers will be hard to represent-even a number like 24 requires at least six numbers: %+%+%+%+%+%-this makes numbers hard to read, and (ii) there is no unique way to write most numbers. We could write $+@, @+%, #+$ or $+#-all for the same number. This too makes it hard to read large numbers.

(3) **Juxtapose old symbols**. Frequently, people want to write $@ instead of $+@-by which they mean "add together dollar and at". This is troublesome for the same reason as (2). Sometimes, people think of $@ as indicating "multiply dollar and at" because of algebraic notation such as "3x" which means "multiply 3 by x".

In the Ordpa number system, we will juxtapose old symbols. But we will not mean either that we should add the symbols, nor that we should multiply them. We will mean something quite different.

In the Ordpa number system, we will write @! for this many things: *****.

We will juxtapose the symbols @ and !. Let's see what the meaning of this is.

@ means we have a single thing (recall it's the first number in our counting sequence). Here, we are using it to indicate that we have a single **group**. And we use ! to indicate that we have no leftover objects.

The way we know that @ represents a *group* in this situation instead of a single object is that there are two digits. Whenever we have a two-digit number, the first digit represents the number of groups and the second digit represents the number of ungrouped objects. Furthermore, we will always make groups of this size: *****.

And we will read "@!" as "flop". So in our new number system, we will make groups of size flop. The table below shows the next few numbers in the system-both how we write and how we say them.

Notice this last number, #!. The # indicates how many groups we can make; the ! indicates that we have no leftovers.

The Ordpa number system is a **place value **system because the meaning of a digit (such as #) depends on where it is located in the number. In @#, # indicates how many ungrouped objects there are. In #!, # indicates how many groups. The place of the digit indicates the value it represents-either a unit or a group of units.

When the @ symbol goes in the flops place, it indicates that we have @ (or 1) composed unit. This new unit (which we call a *flop*) is composed of five units. For more on composed units, read this packet.

Surprisingly, we now have enough information to write any number of any size. We have some basic symbols (!,@,#,$,%). And we have two basic rules for building numbers from these digits-(1) any time we can make a group of flop we will, and (2) we indicate groups by moving the digit one place to the left.

Number language is trickier. It is worth thinking about when we need new words (such as "flop") and when we can use words we already have. This will be the subject of another packet.