What it means to solve

"Do we understand" was on my mind as I was grading intermediate algebra quizzes over lunch today. In particular, I wondered: do my students really understand what it means to solve an equation or inequality? It was clear enough that they knew how to do it, but do they know what they are accomplishing in the process?

One of the learning targets I have identified for my students reads:

I can explain what it means to solve equations, inequalities, and systems, and I can use this knowledge to check my answers for reasonableness and correctness. 

I'll be honest: I really love that learning target. It was inspired based on one of the Common Core State Standards for Mathematics, specifically, one of the Grade 6 Expressions and Equations standards:

CCSS-6.EE.5 [Students will] understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? ... (emphasis added)

That concept is fundamental to the work we do in algebra. Consider the wide array of situations where the sole directive is "solve". Come to think of it, one of my old doctoral advisors once described "solve" as one of the four universal directives of algebra homework sets. (Can you come up with the other three? Answers appear at the end of the post.)

In the case of "solve", the meaning is universal and boils down to "find the set of [thing(s)] that make this [thing] true". Consider: 

• "Solve 3x+4=7" means "Find the values of x that make the equation true."
• "Solve {3x + y = 4, x - 2y = -1}" means "find the ordered pairs that simultaneously make both equations true."
• "Solve y'(x) - y(x) = 0" means "find the family of functions that make the differential equation true."
• "Solve Ax = λx" means "find the real numbers λ and corresponding vectors x, called the eigenvalues and eigenvectors of A, that make the matrix equation true."
• "Solve Ax = 0" means "describe the set of vectors x, called the null space of A, that make the matrix equation true."  

Granted, the devil is in the details, and there are some cases where this interpretation is not quite relevant (e.g. "Solve (9/5)C + 32 = F for C"), but it is a subtle piece of mathematical elegance that this simple concept underlies so many aspects of our work. I wonder: at which point in my studies did I become aware of that commonality of meaning. Hmm.

But back to my intermediate algebra class. To get at my learning target, I included the following question at the end of our most recent quiz:

What does it mean to "solve" a linear inequality? 

I wasn't sure what to expect -- but I suppose that's why I asked the question. I was hoping for responses like this one, which is a paraphrased version of an actual response:

Solving an inequality means finding the set of values (a ray on a number line) that makes it true.

Nice, right? I especially liked the reference to an alternate representation (the ray on the number line). Below are some of the other responses I received, all of which have been paraphrased to protect confidentiality. How would you rate each response in terms of meeting the learning target?

Solving an inequality means:

• ...finding the range of values that allow the inequality to be true.
• ...coming up with a series of numbers that will be true in the equation, not just a set point.
• ...showing how a variable is represented on a number line and the value it holds.
• ...finding the x-value and checking if it is greater or less than the solution you found. 
• ...determining where the two lines meet. 
• ...finding out how one side relates to the other in value.

I see varying levels of understanding in those responses. Some are pretty good, some would benefit from a little polishing, and some suggest an underlying misconception about the goal of our work. It looks like I'll need to watch for opportunities in upcoming unit on quadratic functions to model the meaning of "solve" with my students.

Follow up posts: What (else) it means to solve and Finding wormholes.

(The four universal algebra textbook directives that my advisor used to bemoan are: solve, simplify, factor, and expand.)