A fellow math teacher and writer, Whit Ford, wrote recently about inverse operations. I found his approach interesting and refreshing because he was working on the mathematical relationship between addition and subtraction, and he had set things up in a way that suggested he wanted to look for parallels with other inverse pairs such as multiplication/division and exponentiation/logarithms. Among the principles he cited that I think he’ll do a nice job of generalizing are these:

[Inverse functions can:]

– **not** share convenient properties with the “forward” version of the function, like being commutative or associative

– have ranges that force us to expand the universe of numbers that we had been habitually using before being introduced to the inverse function

– be more challenging to describe verbally than their “forward” sibling

This is lovely stuff which is too often missing in mathematics instruction, or that is at best tangential to instruction. He also suggested three conceptions of subtraction by discussing possible meanings for the expression 5-3:

[1] “five take away three”, or

[2] “what I must add to three to get to five”, or

[3] “five plus a negative three”

In a comment I suggested these additional conceptions:

[4] How much more is 5 than 3?

[5] How far is 5 from 3 on the number line? and

[6] 3 is part of 5; what is the rest?

I’ll expand on these here, and consider how all of this relates to the larger question of inverse functions and operations.

There has been a long-term research project at the University of Wisconsin, Madison called *Cognitively Guided Instruction *(CGI). That project was founded on the principle that if teachers better understand how their students think about mathematics, they will be more effective teachers. One of their early major results was documenting that children have a variety of strategies for solving *addition* and *subtraction* problems before they have been taught addition and subtraction. What is more, the types of situations children encounter in these problems influence what strategies they use more than whether it is formally an *addition* or a *subtraction* problem.

In particular, they identified four major categories of problems based on children’s problem-solving strategies: Join, Separate, Part-Part-Whole and Comparison. Briefly, a *Join* problem involves two sets being joined together; a *Separate* problem involves one set being separated into two sets; a *Part-Part-Whole *problem involves two parts making up a whole, but without any physical action joining them together; a *Compare *problem involves comparing the sizes of two different sets.

Depending on what is known and unknown in the problem, each of these four types encompasses problems that can be solved with subtraction or with addition. The CGI argument is that student strategies correlate closely with these problem types, rather than with the more formal categories of *addition *and *subtraction*.

So subtraction conception [1] above, from Whit’s original list, is a *Separate* problem. [2] is a *Join* problem. [3] is *Join* with negative numbers, which is beyond the scope of the original CGI work.

I wanted to add the other categories: [4] is a *Compare* problem. [5] is beyond the scope of CGI, which worked with set models (e.g. marbles instead of number lines) but is probably most like the *Compare* problems. [6] is *Part-Part-Whole*. Note that [1]-[3] all involve physical actions while [4]-[6] do not. Whether or not there is a physical action in a problem is very real conceptually for young children (and probably beyond!)

Now, from addition we get repeated addition, which is the main conceptual entry point to multiplication. Its inverse is division. Whit’s three generalizations about inverse functions-that they tend not to share convenient properties with their originators that they necessitate new numbers, and that they can be conceptually more challenging, are likely borne out in the multiplication/division example. It would be equally interesting to consider what the major categories of multiplication and division concepts should be. Depending on whom one asks, there can be as few as two categories for division or many, many more.

Next, from repeated multiplication we get exponentiation. Its inverse is logarithm. Once again, Whit’s generalizations apply. But I have not seen anyone try to categorize concepts of exponentiation nor logarithms. Perhaps this is because we have reached some threshold of abstraction in which people only think about these concepts in a formal way. But that seems unlikely-few people think only formally and most people think informally before they can think formally.

And this is the greatest lesson of CGI-if our informal conceptions of addition and subtraction influence how we solve problems at an early age, isn’t also likely that our informal conceptions of other operations influence how we solve problems later on?