On an assessment students did for me today I gave this question:
An aging father left a triangular plot of land to his two children. When the children saw how the land was to be divided in two parts (Triangle ADC and Triangle BDC), one child felt that the division of the land was not fair, while the other was fine with it. What do you think and why? Support your justification with mathematical evidence.
So this student had a hard time with this question. Since there was no height given and the bases were different, she was unable to think about how to compare the areas. She was however able to say that it would be a fair split if the areas were the same. So since I am doing this work this year with giving feedback first and then grades (see past blogpost “Why teachers don’t give feedback before grades and why they should”) I wrote this feedback on the problem set:
I am trying to get her to remember a problem we did in class where there was a similar problem we did with an acute triangle and obtuse triangle that shared the same height:
The area of the shaded triangle is 15. Find the area of the unshaded triangle.
This idea of where the height of obtuse triangles are is a really tough one for some geometry students. But more than that the idea of sharing a height and what effect that has on the area is also difficult.
We will see tomorrow if this student is able to take my feedback and see what whether the division of the land is fair.
By the way, here’s a response that another student had:
Just in case you can’t read it:
“Because the height is the same, it’s the ratio of the bases that would determine which child would get the most land. I think the division of land was not fair, because the heights are the same so therefore the bases are determining the area of the plot. If x=5 then child one would get A=20, child 2 would get 12.5 and that makes the original plot of land 37.5. This means child 2 has a third of the land (12.5:25) (part:part) and half of child 1’s) Even without x=5, the child 2 would only get a third of the land.”
So, all assessments are back to the students, tears have been dried and we are now onto our next problem set (what we are calling these assessments). What we’ve learned is that the rubric allowed us to easily see when a student had good conceptual understanding but perhaps lower skill levels (what we are used to calling “careless mistakes” or worse). We could also quickly see which problems many students had issue with once we compared the rubrics because, for example, problem number 6 was showing up quite often in the 1 row of the conceptual column. This information was really valuable to us. However, one thing we didn’t do was take pictures of all of this information to see if we could have a record of the student growth over the whole year. Perhaps an electronic method of grading – a shared google sheet for each student or something to that effect might be helpful in the future – but not this day (as Aragorn says) – way too much going on right now.
We also changed the rubric a bit for a few reasons. First, we found that when students completed the problem to our expectations on the initial attempt we felt that they should just receive 3’s for the other two categories automatically. We considered not scoring them in this category but numerically felt that it was actually putting students who correctly completed a problem at a disadvantage (giving them fewer overall points in the end). Second, we also changed the idea that if you did not write anything on the revisions you earned 0 points for the revisions columns. Many students told me afterwards that they felt like they just ran out of time on the revisions and actually had read the feedback. This was unfortunate to me since we had spent so long writing up the feedback in the hope that the learning experience would continue while doing revisions.
What we decided to do was to try the revisions this time without the “explanation” part of writing. I think it will keep the students focused on reading the comments and attempting a new solution. I was frankly surprised at how many students stuck to the honor pledge and really did not talk to each other (as they still got the problem wrong the second time around – with feedback). Truly impressive self-control from the students in my classes and how they were sincerely trying to use the experience as a learning opportunity.
I do think the second assessment will go more smoothly as I am better at doing the feedback and the rubric grading. The students are now familiar with what we are looking for and how we will count the revisions and their work during that time. Overall, I am excited about the response we’ve received from the kids and hope that this second time is a little less time-consuming. If not, maybe I’ll just pull my hair out but I’ll probably keep doing this!
First in a series of posts about my experiences with “Feedback Before Grades”
Holy Mackerel is all I have to say – Ok, well, no I have plenty more to say – but after about a week and a half of holing myself up with my colleague, Kristen McVaugh, (big shout-out to Ms McVaugh who is not only teaching PBL for the first time but was willing to dive into this amazing journey of alternative assessment with me this year too), I am totally exhausted, almost blind as a bat, partially jaded and crazy – but mostly ready for a drink. This little looped video of Nathaniel Rateliff and the Night Sweats pretty much sums it up…
So here was our well-intentioned plan: we wanted to start the year off with a different type of assessment. I put out my feelers on twitter and asked around if anyone had a rubric for grading assessments where the teacher first gave only feedback and then allowed students to do revisions and then once the revisions were done the students received a grade. Kristen and I knew a few things:
we wanted to make sure the revisions were done in class
we wanted to make sure the revisions were the students’ own work (tough one)
we wanted to give students feedback that they needed to interpret as helpful so that we weren’t giving them the answer – so that it was still assessing their knowledge the second time around
we wanted to make sure that students were actually learning during the assessment
we wanted students to view the assessment as a learning experience
we wanted students to be rewarded for both conceptual knowledge and their skills in the problem solving too
So we created this rubric Initial Draft of Rubric for Grading. It allowed us to look at the initial conceptual understanding the student came to the problem set with and also the initial skill level. Kristen and I spent hours and hours writing feedback on the students’ papers regarding their errors, good work and what revisions needed to be done in a back-handed sort of way.
Here are some examples:
Student 1 Initial WorkStudent 2s initial workStudent 3 initial work
Some kids’ work warranted more writing and some warranted less. Of course if it was wonderful we just wrote something like, excellent work and perhaps wrote and extension question. The hard part was filling out the rubric. So for example, I’ll take Student 3’s work on problem 6 which is the last one above. Here is the rubric filled out for him:
Student 3’s Rubric
You will notice that I put problem 6 as a 1 for conceptual understanding and a 2 for skill level (in purple). In this problem students were asked to find a non-square quadrilateral with side lengths of sqrt(17). Student 3 was definitely able to find vertices of a quadrilateral, but he was unable to use the PT to find common lengths of sides. I gave him feedback that looking at sqrt(17) as a hypotenuse of a right triangle (as we had done in class) would help a bit and even wrote the PT with 17 as the hypotenuse in the hope of stimulating his memory when he did the revisions.
The day of the revisions Student 3 was only capable of producing this:
Student 3 revisions
He followed my direction and used 4 and 1 (which are two integers that give a hypotenuse of 17, but did not complete the problem by getting all side lengths the same. In fact, conceptually he kind of missed the boat on the fact that the sqrt(17) was supposed to be the side of the quadrilateral altogether.
One success story was Student 2. She also did this problem incorrectly at first by realizing that you could use 4 and 1 as the sides of a right triangle with sqrt(17) as the hypotenuse but never found the coordinates of the vertices for me. I gave her feedback saying there might be an easier way to do this because she needed vertices. However, she was able to produce this:
Student 2s revision
Although she did not give me integer-valued coordinates (which was not required) and she approximated which officially would not really give sqrt(17) lengths it came pretty darn close! I was impressed with the ingenuity and risk-taking that she used and the conceptual knowledge plus the skill-level. Yes, most other kids just used some combination of 1’s and 4’s all the way around but she followed her own thought pattern and did it this way. Kudos to student 2 in my book.
Next time I will talk about some of the lessons we learned, other artifacts from the kids’ work and what we are changing for next time! Oh yeah and some great martini recipes!
