In any endeavor it’s important to know why you’re involved in what you’re doing.  This is no different.  To the unpracticed eye, addition and subtraction problems are pretty much the same, although they may look slightly different from time to time.  However, in my recent study with Liping Ma, I’ve come to learn that these differences are huge and absolutely crucial!

It is important that children are presented a variety of all types of problems, rather than one or two, so that they gain a broad understanding of what addition means and what subtraction means.  Should their only experience with addition be putting two piles of objects together? And should subtraction be limited to taking one portion of the objects away?  No!  Life experiences give us all many variations of both addition and subtraction. Only by giving the children many opportunities to struggle with and experience all types will they be able to build a solid foundation, maintain flexibility with the concepts and go forward in their thinking.

The three types of addition problems and their related subtraction problems are quite different from each other…and they vary immensely in complexity.  Increasing/Decreasing and Combining/Separating problems are all well within the realm of a first grader’s understanding.  Comparing/Examining the Difference problems are far more complex.  They need to be carefully structured and built upon prior knowledge and experience if all children are to be successful in solving such problems with understanding. 

In the beginning, our K/1 team found that most of our children could actually solve comparison problems intuitively, but very few could explain how they solved the problem and express their understanding of it.  We often heard these words, “I just know it, that’s all!” 

We as teachers know that’s not enough.  Children must be able to verbalize their learning, to explain their thinking to their classmates, their parents, and their teachers.  This gives us all a common language and a universal way of communicating with each other in a precise and focused manner.  We as their teachers can ask age-appropriate clarifying questions in a nonthreatening way for our students to grapple with.  We can also address misunderstandings by posing questions that lead their thinking in more directions.  In all our work, we must strive to provide a nurturing environment in which it is safe to take risks, for it is here where the giant leaps of understanding occur. 

What a joy it is to observe the children’s steps along the way as they construct their own understanding of just what addition and subtraction mean!