Author Archives: Everett Gaston

Composite Units in Learning Math

The term composite unit is important in understanding how children come to know math. When a student builds math concepts with a composite unit as part of the foundation, there is a real perceivable difference in what the student can do compared to students who learn math simply as a set of procedures.

One of the early uses of composite units for students is in doing addition and subtraction. Give a student the basic problem: 22 – 17 =  __. “School math” would have a procedure or algorithm utilizing borrowing. I use the term “school math” to describe what happens in a traditional classroom where a procedure is initiated by adult understanding rather than by the student’s mathematics. The student could follow the procedure and even get the correct answer, without really understanding what happened or why. With an understanding of composites, the student would understand that 17 is contained in 22, and the goal is to find the remaining part. The student could then just count from 17 up to 22, monitoring the count to get the answer. While this may sound more primitive, a better, deeper understanding will be the pay off in the long run. The important thing to know is that this understanding of counting-on cannot be taught. The procedure of counting-on can be taught, but it must be the initiative of the student for it to have meaning.

My team has been working on understanding the composite units for multiplication, using what we refer to as towers tasks. Multiplication is the coordination of two composite units so that one of the composite units is distributed over the elements of the other composite unit. The towers tasks, created by Ron Tzur and others, are designed to help students learn this in the context of a classroom activity. We have confirmed that students utilizing the towers tasks come to have a clearer understanding of what they are doing by multiplying numbers rather than students who are just memorizing the multiplication table.

The next step for students would be to develop fractions based on fractional composite units. It would follow naturally from what has already been learned and allow for more understanding than the empty procedures that have become the basis of “school mathematics,” especially with fractions. For example, take a fraction like 1/5, and treat it as a unit, the way we treat 1. In “school math,” 1/5 is usually thought of as 1 part in 5, and is not usually treated as a unit to count or measure. The traditional approach is limited, in that it doesn’t allow students to learn to work with fractions in the same ways that they learn to work with whole numbers.

Starting with a fraction as a composite unit, other ideas naturally fall into place, like counting and adding. For example, how many 1/5’s are contained in 3? By the traditional “school math” method, a student would be taught to divide 3 by 1/5, which involves the division algorithm. Students might get a better grounding by simply counting by 1/5’s up to 3.   

Following learning fractions, with just a change in the composite unit, the students can move on to the next step. By calling x (an unknown quantity) the composite unit, and having available all the same computing schemes and operations, Algebra is a natural and understandable development.

Can you think back to learning math in school? Was there ever anything that didn’t seem to make sense, yet you could do the procedure well enough to solve the problems?


Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4(3), 259–309.

Tzur, R., Johnson, H., McClintock, E., Xin, Y. P., Si, L., Woodward, C. H., & Jin, X. (2013). Distinguishing schemes and tasks in children’s development of multiplicative reasoning. PNA, 7(3), 85–101.


NCTM Reflection

We recently attended the annual meeting of the National Council of Teachers of Mathematics, NCTM. This year’s conference was in San Antonio, Texas. We were mostly a group of research associates from the AIMS Center. The size of the conference (over 7000 attendees) and the wide range of presentation topics worked together to create a… Continue Reading

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,… 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


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

Coordinating Units, Part 3: Fractions

When looking at coordinating units, it is important to consider concepts other than just multiplying. One of those is fractions. Fractions would be first among these encountered by a child in school. Mental operations that must develop for a child to understand fractional ideas include partitioning, iterating, and splitting. These developments are not taught, much… Continue Reading

Coordinating Units, Part 2

This post continues my September 20, 2016 post, “Coordinating Units: A Brief Introduction”. Last time, I introduced a problem to illustrate the basic differences between additive reasoning and multiplicative reasoning used to solve a problem. I also defined levels of units and what it means to “coordinate units”.  In addition, I said that “in activity”… Continue Reading