In earlier posts, I discussed the "In the cups" performance task and my use of a proficiency-based assessment system in my intermediate algebra course. In this post, I'll share an example to illustrate how the use of an evidence-based rubric has supported my implementation.
Here's the rubric, which is adapted from one that my colleague Math Hombre used when he was teaching the same course. The rubric is focused on the quality of evidence provided for proficiency with a given learning target.
0 (F) No Evidence / Missed Opportunity. There is no evidence of this target available yet.One of the prompts from the cups task is:
1 (D) You’re not there yet. The evidence suggests you need additional support, you may have some misconceptions to overcome, or both.
2 (C) I’m not convinced: the evidence is mixed or not yet convincing. Sometimes you make good progress, but you also make errors, get stuck, or struggle to complete some tasks.
3 (B) I’m almost convinced: You can probably do this, but I don’t know if you can do it consistently.
4 (A) I’m convinced: There’s good evidence that you can do this consistently. You explain your reasoning clearly, and any mistakes tend to be minor and easily corrected or explained.
5 (A+) I’m sold: You obviously “own this”: you understand it in a deep way, rarely make mistakes, and communicate your understanding clearly and convincingly.
"How would you convince a skeptic that your model is correct or reasonable?"It is aligned with this Learning Target:
___1.11. 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.Here is one student's paraphrased response on her first submission of the cups task:
"I would measure the heights of different stacks of cups and comparing the values predicted by the formula."I'm almost convinced she can do what the learning target says, and I'd hate to "take points off" (deficit model), but it really would have been much more convincing had she actually made a measurement and tested her mathematical model against it.
Thankfully, I'm not scoring with points anymore.
So I wrote: "Can you show me? You have the picture..." and recorded a rubric score of 3 ("I'm almost convinced") for Learning Target 1.11. She has a 3, which translates to a "B", and she could quit now if she wanted to. Interestingly, my observation has been that when I give feedback with a rubric instead of "taking away points", my students are much more likely to resubmit the task with an improved response.
This student did indeed resubmit the task a few days later. Here is her updated response (again, paraphrased):
From the picture, we have two known points. To check the model, you could substitute the # of cups into the model and calculate the height to see if it matches the picture.The evidence of her proficiency is much more convincing now. What rubric score would you give this response?
When we do that for 1 cup, we have y = 0.49(1) + 6.77 = 7.26 cm. The picture shows 7.2 cm. When we do that for 20 cups, we have y = 0.49(27) + 6.77 = 20 cm, which is just what the picture shows.
Another prompt from the cups task is:
"How would you use this context to discuss the slope as a unit rate of change?"It is aligned with this Learning Target:
___ 1.8. I can identify, describe, and interpret the slope of a linear function in context.One student's response on her first draft:
The slope is 0.49. Slope means rise over run, so it means 0.49 up the y-axis and over 1 on the x-axis.Again, I'm almost convinced, but she has not discussed the slope in the context of the cups task. So I circled the last two words of Learning Target 1.8 and wrote: "How would you relate it back to the cups?"
Her next draft was much improved:
The slope is 0.49. As a unit rate of change, the slope tells us how much the y-value changes when x increases by one unit. In this case, it means the stack height increases by 0.49 cm each time 1 new cup is added.Now there's a convincing display of proficiency!