I can show you more than that

In a previous post, I talked about my grading system based on the mantra, "Show me what you can do." Here's a quick example of a student's response to a quiz item that shows why this proficiency-based approach feels so right:

One of my learning targets was:

___1.5. I can solve linear systems and represent the solution symbolically and graphically.

I stated that objective at the top of the quiz, along with several others, then gave the following task:

The solutions to the following system of equations are provided. Show that you can use the elimination and substitution methods (use each one once) to solve these problems.

{y = 3x+6
{2x + 4y = -4              solution is (-2,0)  

{7.5x - y = 10
{15x - 4y = 10            solution is (2,5)

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."