Unit Coordination

Intensive Quantity and Extensive Quantity

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?

References

“Intensive Quantity and Referent Transforming Arithmetic Operations”, Judah Schwartz “On the Operations that Generate Intensive Quantity”, L. Steffe, D. Liss, H. Lee

An Update

Things are really hopping around the AIMS Center. Everyday becomes better than the last. I wake up and I’m challenged and excited by what I get to do during the day. As most people know, we really want to find a way to share what research tells us about children’s construction of number with classroom… Continue Reading

Composite Units

Counting-on is one of the things I have come across in Les Steffe’s research that is crucial, but not necessarily an obvious goal to have for students. It would seem that if a student could count-on (ex: given the problem 6+5, would start at six and count-on five more rather than starting from 1 and… Continue Reading

Toward Number: Hiding Counters

In my last blog, we discussed how the student needs time to imagine counters, or use something that can stand in the place of counters, so the child will gain enough experiences to make just the numeral meaningful. How can we encourage students to do this? Let’s imagine a child has the goal of figuring… Continue Reading

Reifying Math

In school mathematics, we spend a lot of time making math very formal, very sophisticated, and very unreachable for most people because it doesn’t feel real. Perhaps more time should be spent playing with math, exploring math, and making math real for everyone. In ancient times, people often did very sophisticated math problems, but they… Continue Reading

Partitioning

When you hear the term partitioning, you might think about partitive division or partitioning a discrete set of objects, like dividing a dozen cookies among four people. Partitioning also applies to continuous intervals. An example would be the task of equally sharing a candy bar among 5 friends. The outcomes for how a child would… Continue Reading

Rich Questions

This month kicked off with a bang! It started with a trip to the beautiful Monterey Peninsula for the California Mathematics Council (CMC), North Section – Asilomar Conference 2016. If you have not been before, switch over to your calendar right now and mark off the first weekend of December for the conference. Then you… Continue Reading