Wednesday, October 10, 2012

"Substituting for x" is a subtle killer

"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.


  1. I'm running into this now, in addition to solving basic equations.

    I'm in my second year teaching and until now (including student teaching) I had not taught Algebra I. Everything up to Calculus, but it's my first time teaching Algebra I (to both 8th and 9th graders). It's little, subtle topics like this that I'm finding most difficult. It is very hard to transition myself to what they are and are not comfortable with in both arithmetic and variables when I've never had to explain what "evaluate" means. I like the insight that a variable is just a number waiting to be revealed.

    Function notation is one of those things I see as a big hurdle coming up.

  2. I have the theory that math is about three completely different skills: arithmetic, geometry, and logic. These are based I think on different parts of the human brain. Disclaimer: I cannot prove that scientifically.

    Manipulation of numbers is only arithmetic. To develop the idea that an operation combines two numbers to form a third is geometric. And logic is needed to realize that (a+b)+c is the same as a+(b+c) and thus to organize the computation.

  3. I can't tell if my post helps or hinders your post, but I felt my grade eights hit it out of the park when I introduced this this lesson .

    My focus was to get them to understand that variables mean ANY number. Different variables give different outputs.

    I only did it with basic equations however, we will see what happens with simplification and exponents... (eek).

    1. First: I love the post.

      Going a bit deeper, how many mental steps are involved in evaluating an expression (or a function)? In a poll of 100 teachers, I'd expect 95 to say that there are these two: (1) Replace the variable with a number (2) Do some arithmetic.

      What I'm pushing for is a different story about evaluating expressions:
      (1) Translate the expression into a rule
      (2) Apply the rule to a number

      So rather than encouraging a "fill in the blank" model, I'd prefer to get at the "any number" understanding by having students apply some sort of process to various numbers, and summarize that process in terms of an expression. At the Alg1 level I talk about number tricks. At the Alg2 level I talk about rules.

    2. I want to hear you elaborate what you mean by rule? How is rule different than the arithmetic?

      My thought over the last years has been focused on two models.

      1) A variable is any number.
      2) A variable is a mathematical object.

      Students really got into the choosing their own variable in my mad lib lesson. They could do it very easily. When they did the personal practice and someone (Pre-algebra with Pizazz! yippee!) else picked for them the value of x they were stuck.

      My example was 5x if x=3. I walk through students by saying, "We have five x's." I literally drew five x's on their paper.

      X X X X X

      "If those are each equal to 3, how many do we have?"

      "We have 15?"

      "How did you figure that out?"

      "I multiplied!"

      I am doing my best to prep them for distributive property and multiplying monomials by engaging very concrete understandings of variables as objects, but in the mathematical world (very Platonic, I know but it makes sense to me). I don't know if I have missed the mark, or if I am doing things wrong, but it has made most sense to me in this way.

    3. I'm sure you haven't missed the mark. (BTW, were the kids confused by the "5x" notation?)

      What I'm trying to describe is the difference between calculating "(5 + 3) * 9" and applying the rule "add 3, then multiply by 9" to the number 5. Though, of course, both students would end up doing some arithmetic, there's a mental step before arithmetic in the approach I'd like to foster. I'd like my students to be aware of applying the generalized rule "take your number, add 3, multiply by 9" even as they're doing particular calculations.

    4. Hear is a post involving "Number Tricks" highlighting a student error that I believe addresses your point: