Teachers, parents, and students each need a framework of guiding principles to organize the teaching and learning of elementary mathematics. Several powerful principles provide a practical framework within which you can fit mathematical procedures and concepts. This article will examine the principle that "quantities have names."
Much of mathematics is counting - and knowing what we are counting. Like people, quantities have first names and last names. The first name tells us how many, and the last name tells us what kind. For example, comparing the expression 4x with 400, the first name of 4x is 4 and its last name is x, while 400's first name is also 4 but its last name is hundred. This connects with many real-world situations with which children are familiar: hold up all the fingers on one hand and ask a child "say what you see," and they will properly say, "Five fingers." Five is the first name (how many) and fingers is the last name (what they are).
This first name/last name principle has two natural connections. First, children often have friends or classmates who share the same first name, so they are familiar with sometimes having to refer to both the first and last names of someone in order to clearly identify that person. Second, students are teaming to read text from left to right, top to bottom on the page. The name for the mathematical term "4x" is verbalized and written just like we read a person's name - left to right, first name then last name.
It is important to connect this naming principle with the important "exchange" notion that is a central part of place value. We can extend this principle to explain that quantities sometimes have more than one name. For example, when I am in the classroom, students address me as "Mr. Hazen." When I am at home, my sons address me as "Dad" and my wife calls me "Bob" (or even "honey"). The point is that whatever my name is, I am still the same person. Likewise, the quantity "40" has more than one name. We can call it four tens or we can call it forty units. The quantity 345 also has several names, among them being three hundreds, four tens, five units, or three hundreds, forty-five units, or three hundred forty-five units.
This notion of multiple equivalent names emphasizes that in place value, the last name comes from the position that the digit is placed in. Thus, 400 is four hundred because the digit 4 was in the hundreds' position, while 40 is four tens because the digit 4 is in the tens' position, and 4 is four units (or four ones) because the digit 4 is in the units' (or ones') place. Knowing that 40 can be thought of as either four tens or as forty units is extremely helpful in developing number sense and the ability to decompose and recompose numbers. Knowing that the last name for a base ten number is determined by where the digit is placed is a set-up for success when students begin working with decimal numbers such as 0.4 and 0.004.
This first name/last name principle can also be used to explain why the whole number 7 is read simply "seven." Certain people are so famous that we only need to use their first name to identify them. In Minnesota where I live, one of the most popular people is the baseball player Kirby Puckett. He's so popular and well known that when people mention Kirby, we usually know that they are talking about the baseball player, even though his last name wasn't said. So Kirby and Kirby Puckett usually refer to the same person. Does 7 have a last name? Well, it actually does; its proper full name is seven units. However, we run into the whole number 7 so often - 7 is so famous - that we just call it by its first name only - seven. So seven and seven units usually refer to the same quantity.
This first name/last name principle applies to almost every area of mathematics, such as:
- Decimals. For 0.04, we read four hundredths, where the first name is how many, four and the last name is what place it is - in this case, the final decimal place, hundredths.
- Fractions. For 4/7 in vertical form, its name is read top to bottom, first name then last; the four tells us how many, and the sevenths tells us what kind (sevenths).
- Algebra. In algebra, the first name/last name principle is a "set-up for success" when students formally study algebra in junior high school. The reason why "4x + 3x" can be simplified to "7x" is because they have the same last name. In contrast, "4x + 5y" cannot be simplified, because their last names are different. In later years of formal algebra, students will realize that "same last name" is an informal definition of "like term."
- Geometry. 40 pi is read forty degrees, so its first name is forty and its last name is degrees.
- Measurement. 4 units of metric linear distance would be four meters. 4 units of metric area would be four square meters (with the last name being square meters). 4 units of metric volume would be four cubic meters (with the last name being cubic meters). 4 units of time would be four hours.
This is only a partial list; you can add your own first name/last name quantities as you work in almost every area of mathematics. The principle that quantities have first and last names is a powerful tool that helps students organize their thinking about mathematical facts and concepts in several ways. It fosters connections between mathematical strands such as arithmetic and algebra; it shows similarity between integers, decimals, fractions, variables, and angle measures; and it reminds students that mathematics is a language that makes sense.