#11 Defining Place Value

As part of the Molalla Math Project, we put together a definition of the place value system of numeration that I want to summarize here. Place value is a rich system of mathematical operations for creating and transforming numerals such as borrowing, carrying, column-based computions, algorithms, and so on. It’s a basic and very important math idea, but unfortunately students encounter considerable difficulty in understanding it. Our research showed that by middle school, only about a fourth have a handle on the concept.

So it seems important that before we do anything, we be clear about exactly what this thing is we call place value. We’ve got to understand the nature of this numeral system with its particular operations that is distinct from the whole number system and its operations. Without this precision in our own understanding of place value, it seems unlikely we can evaluate and work with students’ place value understandings and misunderstandings, or conduct objective research into the development of students’ understanding of place value.

But we found something strange. When we turned to expert sources that we would expect to provide us with a clear description of the place value system itself, we could not find help from any of the math textbooks and teachers’ manuals, the state curriculum objectives or materials, or even the math education literature we searched. All we could find about place value were the common procedures everyone uses with place value but not definitions or explanations of the place value system itself. It appears to us that the math ed field thinks that is all there is to place value. The filed describes nothing but various procedures and problems involving place value but not the place value system of numeration itself. If the experts in the field take for granted any definitions and explanations of the place value system and refer only to calculation problems and directions for performing various procedures and algorithms, then it is seems likely that students’ knowledge of place value will be reduced to only whether or not they can perform these place value procedures and problems correctly to produce correct answers. Without knowing what understanding the system of place value is, of course teachers and students must rely on memorization of place value procedures.

Obviously if we don’t know what place value is, then we can only work with students’ behaviors and perfomances in following place value procedures and not students’ understanding. If we cannot know whether students understand place value then obviously we cannot teach for understanding. So teaching and research both appear to be based on getting students to follow and remember procedures rather than the underlying place value system itself. So we need to be honest with ourselves. How can we teach place value, insure students understand it, or conduct research into its psychological development in student thought if we don’t have some precise definitions of what place value numeration is?

Since we could not find help from the math ed field, we decided to tackle the problem ourselves. We began by thinking about how mathematicians and scientists define things, and in particular, math systems. One approach we found was listing the necessary and sufficient conditions. These properties are important but necessary and sufficient conditions define valid properties of place value system but not its transformations themselves. A definition of a math system has to include more than properties. It has to include the transformations as well. It’s the overall system we’re after, not just its properties.

We find that the more formal methods by mathematicians specify both the elements of the system and the transformations of the system. We could then use this idea to describe the elements of the system and then describe how its operations work. The operations operate on the elements to add them, subtract them, multiply and divide them. So we have a method for defining place value system, define its elements and then define its operations.

The elements of place value system are the numerals. Numerals turn out to be a bit more complicated than they first appear. The numeral 16 appears to be two digits that represent sixteen counters. But numerals actually have some key aspects that necessary if numerals are to work properly in a place value system. We listed six aspects that make up numerals.

Place Value Numerals

First, numerals are composed of one or more digits placed together; these are the digits 0-9 of our common place value system.

Second, numerals use the places as signaling indicators. The ordinal position of the places occupied by the digits stand for certain place values. The ordinal position is a representational element indicating the place value the digit is counting. The one in 16 is a count of the base value of ten, that is, a count of the number of groups of 9+1 elements.

Third, numerals have a constant base that is the fixed size for any group that can be formed. The group size is not usually explicitly represented unless we move into another base other than our standard 9+1 grouping. Groups are either of single elements or other smaller groups but any grouping is always exactly the size of the base; it is a constant. In the standard system, the base value is always 9 + 1. When we count one more than 9, we always call the next number of things 10 or one group. The base value of what we call ten (one group) is actually one more than 9. So we need only enough unique digits to count up to the size of the group or base since then we start over by counting whole groups until we get to 9 and start over again counting groups of groups. A place value system of numeration always has one less digit than the standard size of the base, plus it has the digit 0.

Place Value Operations

We use five forms of place value operations in organizing numerals. We use two operations inside a single place and one inside a single numeral. We also have two operations that operate across places or numerals to move values among places.

First, inside a place, the repeated multiplication of the base represents a place’s value. To determine how many times the base is multiplied times itself, we use the ordinal value of the place’s position minus one. Thus the place value corresponds to the placed because both the base the and position are involved in composing it. For the first position, we use the base 0 times for a place value of 1. For the second position we use the base once for a place value of 10. For the third we multiply the base value times itself twice for a place value of 100. (The more advanced way to represent the repeated multiplication of a single value is by exponents. The first position is 10 to the 0 power or one. The second position is 10 to the second power or ten squared and so on.)

Second, inside a place, the multiplication of the digit in the place times the place’s value represent the place’s total. In 26, for the tens place there is a total of two tens or twenty.

Third, inside the numeral we combine, add, each place total to form the numeral total. In 26 we add the place totals, the two tens and the six ones, to form the numeral total.

Fourth, we have operations among the places that move values up the places such as from the ones place to the tens place. We use this operation of moving value up the places in the form of regrouping such as when we combine two numerals in column addition, e.g., 26 and 16. The total value of the two numerals combined remains constant after the regrouping so that after adding up the ones to get the total of 12 ones and 3 tens, we can regroup to get 4 tens and 2 ones without changing the overall value.

Fifth, besides regrouping which moves values up the places, there is its reverse operation that moves values down the places as in borrowing.

The Place Value System of Numeration

The place value elements and operations form a valid system, that is, a coherent whole within which we can use the operations to transform the elements "move around" the elements without contradiction. We can change a numeral by moving values up or down the places by "borrowing" or "regrouping" in any way and as many times we like and the overall total value will remain constant. We can exactly undo every change we make by reversing the borrowing operation with the regrouping operation. These two operations are the exact reverse of each other and the produce a null or zero change when combined. Furthermore, the order in which we do these operations does not matter, the end result will be the same. And we can use 0 in a place as a null or no change operation when we combine it with other place totals. Thus the place value system provides us with all kinds of operations we can use to transform a numeral while keeping the overall total constant. The place value operations produce an infinite number of numerals each of which is always another numeral. The system has a rich, mathematical logic.

The System Itself

This rich mathematical logic having the kinds of properties we described tell us the place value system is truly a mathematical type of system and not just a set of symbols or a language or figural representations for number. It is a distinct system with its own operations and elements that are not to be confused with the number system. Place value numerals are generated mathematically; they have a mathematical composition, not a figural one like words. These properties of place value make it a kind of math system we call a mathematical group. A mathematical system with group properties have a direct and reverse operation, associativity among its operations, a null operation, and closure such that the operations always produce some new numeral.

Conclusion

Now that we see the rather complicated make-up of the parts of a place value numeral and the mathematics of the place value operations, it makes a little more sense why this mathematical richness causes 3/4s of our students to have so much difficulty understanding place value.