In discussing coordinating units as a way to understand multiplicative reasoning, it is not always evident that there are differences in multiplicative and additive reasoning. What I want to do is give some examples to help clarify the differences.
Multiplication is often presented to children as repeated addition. But there is more. In math classes, we ask students to operate on numbers with little regard to any real application, and we don’t usually put any units of measure in problems. This simplifies the “procedure”, but leaves out what is needed for making connections to real applications. I bring in the units of measures, which leads to two types of quantities: intensive and extensive.
An extensive quantity can be thought of as one that changes when the amount of the measured substance changes. Some extensive measures would include volume, length, and mass. Two extensive quantities of the same units are additive. For example, 5 inches + 8 Inches = 13 inches or 4 cups + 3 cups = 7 cups. The adult (teacher/parent) would likely emphasize the numerical part of each. It is important to note that it is an important step in cognitive development for a child to understand measure.
Intensive quantity doesn’t change by having more or less of the substance. Some examples of intensive quantity are density, speed, and molecular weight. The density of some substance doesn’t change whether you have a lot or a little to work with. Density is mass divided by volume, the ratio of two extensive quantities. Speed is distance or length divided by time. This type of unit is the result of, or used in, multiplicative reasoning situations. For example, (60 miles/hours) x (3 hours) = 180 miles.
The three quantities in the last example all have different units of measure. It would not make sense to add the quantities. But working without units, a teacher might say that 3 x 60 = 60+60+60. While that is true as a math fact, what happens to the units, when a student applies math in context? What types of extensive and intensive problems do you find in your curriculum?
“Intensive Quantity and Referent Transforming Arithmetic Operations”, Judah Schwartz “On the Operations that Generate Intensive Quantity”, L. Steffe, D. Liss, H. Lee