"So I just swap the number and then treat it like arithmetic? Oh, that's easy!"
Here are some common mistakes
kids make when evaluating expressions or functions:
- Able to evaluate forward, but unable to undo the evaluation, even when given something like f(230).
- They'll swear to you that a^2 is -1 when a is -1, because -1^2 is 1, even though they know that (-1)^2 = 1, and that -1 times -1 is 1.
- They'll make weird calculation errors when evaluating expressions that they wouldn't make if they were just doing the arithmetic.
I think that if you want to help your kids avoid these mistakes, you're not doing them any favors by talking about swapping, replacing, substituting or blanks. All of this language support a "mystery value" picture of expressions and functions, where variables stand for particular numbers, and every variable is just waiting to be revealed as standing for a particular mystery number.
Instead, it's helpful for kids to think of expressions and functions as operations to be done on any number. Number tricks
are a nice way of setting this up, but I think that you can undercut things by talking about swapping/replacing/blanks when dealing with expressions or functions. The reason is (and this is subtle, and possibly wrong) because swapping says "this expression is just about particular numbers."
Better language would be applying the expression/function to a number. This emphasizes that the expressions says something about numbers in general, which can be applied to any particular number. (Evaluating is fairly neutral language, but not if you define evaluating as "substituting.")
There's a further difficulty when teaching function notation that I want to get off my chest. If you introduce function notation with evaluation, and define evaluation as swapping, kids miss out on the subtleties of the notation. Why do they miss out? Because evaluation with swapping is too easy -- you just ignore the random letter before the parentheses, take the number inside the parentheses and swap any variables with that number.
But does that f stand for something? And what are the parentheses doing? What is f equal to? What if you have an f inside the f
? Is that like f times f? And what does this have to do with outputs and inputs
? Does f stand for the output?
Evaluating functions with swapping doesn't give kids enough friction to force them to notice the weirdness of this notation. And that means that they're missing out on the move from seeing functions as processes to seeing them as mathematical objects
, the sorts of things that we can use adjectives and predicates to describe.