Let's examine a typical Algebra 1 course. Here's the Intro page for Chapter One.

When m=4 and n=5, what is 3*n + m ?

and 34 - 3/11 = ?

This is fifth and sixth grade stuff. I expect all my students to be able to correctly evaluate these two expressions in a 9th grade algebra 1 class, but I know that there are always kids who mess up, need reminders, or who just don't realize that High School is different and paying attention to your work matters.

The point is that everyone gets an "A" here, or 100% if that's your school grading system.

Fast forward to chapter 6, linear systems. There are many ways to solve systems and we expect that each student learns them all. Invariably, there's a problem.

Johnny gets half of them wrong all the time. He gets a 50%. If you look closer, you see that he can only answer the most simplistic questions and then only occasionally. If you were pressed, you'd admit that he didn't understand at all. He can't explain what he's doing and didn't show any work, you suspect that he copied many of his answers from nearby papers, but you can't prove it so you give him a 50%.

No problem, he says, that's a 75% average. And, if you include his 100% homework grade and 95% class participation grade, that's easily a B+.

## How does understanding one aspect of arithmetic average with not understanding algebra to get a "Good", almost "Excellent" grade? |

Hopefully, we can do that with PBE.

Each course is defined as a set of Big Ideas. In order to get credit for the course, each student must understand all of them.

That's a huge shift.

Think about it, though.

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