If a function is a vending machine, then…

A one-to-one function

A function that is not one-to-one

Constant function

Identity function

Not a function

Inverse operations

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?

CAUTION!

A textbook I am currently using in my College Algebra course has a number of boxes with notes to students like this one:

“CAUTION! Clearing fractions is a valid procedure when solving rational equations but not when adding, subtracting, multiplying or dividing rational expressions.”

The remainder of this cautionary box goes on to clarify the meaning of this statement, but does nothing to explain why we use one procedure when solving and a different procedure when adding, etc.

Why not?

Are we worried that the real mathematical story is too hard for College Algebra students? I don’t think that it is (and more on this shortly). Do we believe that this represents the true nature of mathematics? I know that it is not.

Mathematics is not an arbitrary set of unrelated procedures. It is a coherent body of knowledge that includes a number of really useful procedures. But when these procedures dominate the public face of the subject, there remains nothing left to connect them to each other and we have to issue all sorts of cautions and warnings to our students so that they will remember which procedures to use in which situations.

My College Algebra students (pity them) have been privy to my first attempts at building these sorts of connections. So here is the real mathematical story as I see it…

Rational expressions are things like this: (x+1)/(x-3). They have a polynomial in the numerator and one in the denominator. Rational equations are things like this: (x+1)/(x-3)=2. They have rational expressions in them and an equal sign, so one thing is equal to another. Often we want to solve rational equations, which means that we want to find the value(s) of x that make the equation true. One procedure for doing so is commonly referred to as “clearing fractions”. To clear fractions, we multiply both sides of the equation by the least common denominator of the fractions involved. In the example, there is only one denominator (x-3), so we multiply both sides by this. After simplifying, the result is (x+1)=2(x-3). This is now a linear equation in one variable and solving it is a well-practiced skill for College Algebra students.

But what is going on when we clear fractions? We maintain equality by multiplying by the same expression on both sides of the equal sign, but what’s really going on? Consider an alternate way of thinking about solving the equation (x+1)/(x-3)=2. An equivalent form of the equation is, (x+1)/(x-3)-2=0. If we think about the left-hand side as a function, f(x)=(x+1)/(x-3)-2, then we are trying to find the value of x that makes the function equal to zero. We are trying to find the roots of the f(x)-the places where the graph of f(x) crosses the x-axis.

When we multiply both sides of the equation (x+1)/(x-3)-2=0 by (x-3), we get this: (x+1)-2(x-3)=0. On the left-hand side we have “cleared” the fraction and on the right-hand side 0 times anything equals zero. Notice that if we think of the left-hand side as a function it is a different one, g(x)=(x+1)-2(x-3). We are looking for the roots of a different function. But g(x) has something important in common with f(x). They have the same roots. In the graph below, f(x) is in blue and g(x) is in red.

Indeed, we can multiply a function by any meaningful value or expression (the various forms of 1/0 do not count as meaningful!) and we will get a new function, but we can be sure that the roots of both functions will coincide because anything times zero is equal to zero.

So why can we clear fractions when we are solving equations but not when we are simplifying expressions? Because when we multiply by the least common denominator we are changing the function. The new function has the same roots so the new equation has the same solutions. But the new function has a different, and non-equivalent expression.

There are other ways to think about the relationship between these procedures, of course. I do not mean to advocate for this function-based approach as the solution to a complex problem of teaching and learning. But I do believe that operations on functions (which are a standard topic in College Algebra courses anyway) can provide reasons for these procedures, and that these reasons are within the grasp of College Algebra students. I believe my students can learn to think mathematically. CAUTION boxes imply that they cannot.