Scoring an Ordered List

My 7th graders have a question on their exam that asks them to put eight numbers (integers and fractions) in order of their distance from 0 on the number line, starting with the smallest distance.

These types of questions are tricky for me to grade, and because there are eight numbers in this sequence, the task of grading it fairly suddenly becomes thorny and irksome.

Let's change the question to this:

Put these numbers in order from least to greatest: 5, 7, 2, 3, 1, 4, 6, 8

The correct order is 1, 2, 3, 4, 5, 6, 7, 8 :) — for a possible score of 8 points. How many points would this response earn?

1, 2, 4, 5, 3, 7, 8, 6

So, only the first two numbers — 1 and 2 — are placed correctly. Is the score just 2 out of 8 then? But I want to give some credit to 4 and 5 being next to each other, likewise with 7 and 8.

I've tried to come up with some metric to score this, and then I would want to apply the same metric to different sequences to see if any would break my invisible "fairness" barometer. For example, whatever score I came up with for the above sequence, I think the below sequence should get a lower score because the 7 and 8 are farther upstream than they should be.

Anyway, I have some ideas. The above two sets are Sets A and B below.

I wonder if there's a way to score an ordered list that half of us math teachers can agree upon. I'd like for my students to think about this too. Meanwhile, here is a spreadsheet with my scores if you'd like to take a look and play along. Just enter your name in row 1 (and link your name to your Twitter, if you want) and the scores you'd assign to these sets.

[02/01/18: @MrHonner had a similar question over 4 years ago: Order These Things From Least to Greatest.)

New Marking Strategy

When grading a 10-point assignment, I have a hard time deciding if the work shown is worth 4, 5, 6, or 7 points. If I like the kid, then I’m giving her a 9. If I’m hungry, then the kids gets a 4. If the kid’s mother gave me a $25 Starbucks gift card, then the kid gets an 8 with a drawn smiley face.

What I end up doing is giving every less-than-complete paper a 1 — yes, ONE — and so far this marking strategy seems to be working.

I want kids to revise their work until it’s flushed with coherent mathematics. A score of only 1 at the top of their paper — along with my comments — motivates them pisses them off so they go back and revise their work. If they need help with the revision, they know where to find me. If the revised work is still not up to par, then the score stays as a 1, and the kid gets to revise it again and again. Or when the grading period ends. And while I haven’t kept track of any hard data, I’m willing to bet that the revision rate has at least doubled.

Grading papers sucks. But grading with a 1 or 10 has alleviated much of the stress. Like I found a cure for my crazies.