The best lesson I've ever taught? (Part 2)

​Using student feedback, problem solving, and modeling to come to a better understanding of assessment

Continued from this previous post
So, I let the issue marinate a bit, had the same discussions with my other two classes the next day (block scheduling), and formulated the following situation. I took a break from the curriculum and devoted the next two days of classes (5 different sections, 90 minute classes) to investigating in the hopes that students might come to a greater understanding of how their grades are calculated, and maybe even learn how weighted averages work in the process.

Theory
As I mentioned in the previous post, I wrote this shortly after a workshop I attended (the first of five institutes for the Math Fellows in International Schools program). Whether consciously or not (again, this was way back in October), the discussions we had on mathematical modeling really informed the writing of this problem. Previously, I would have given some numbers, but here I required students to simply use my grading scale, and come up with their own interpretations of intentionally vague descriptors such as "or slightly better" or "just squeaking by". 
​I didn't (still don't) have a great understanding of "mathematical modeling", and found (still find) it difficult to include in the work I ask students to do. This problem feels like my first decent attempt at formulating a modeling situation.

Practice

Everyone had trouble getting started
"You want me to make up my own numbers?"
- "Yup, just make sure they make sense"

Most of the students started off with their "feelings" about the problem, which is great. I think we want students to speculate about the solutions before they get into the math of it.
However, many of them, particularly the older students, were weirdly overconfident and considered themselves "done", without doing the math to confirm.
​The images above show work the students asked me to check, to see if they were right... 

We ended up going through Tariq's story as a class, to make sure we all understood the math, then assigning groups to each of the others to get through all of the students in one class period.

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Timing was a real challenge with this, as I was presenting the same problem to 5 very different classes. Nothing that worked for my first two classes worked for my third, etc. Ended up making a table like this for most of them, and having students fill them in as they figured things out.

The Math
The model that ended up making the most sense to the most students (which I didn't get a picture of, unfortunately) was each category of grades (practice work, summative assessments) being a bowl full of available points. You earn a percentage of those points based on your average in that category.

These guys had an "aha" moment when they figured out the total is 95, not 100 (I have 5% in reserve for a category we don't use yet). 
Further, if I don't grade practice work, it's out of 85...

Conclusion(?)
So, the whole point of doing this was to really investigate the question:

In what case does grading practice work further my ultimate goal, which is to make grades reflect the level of understanding of my students?

Working through this problem allowed most of the students (with a little help from me) to come to the following (seemingly obvious, but important to understand) general conclusion: grading practice work is only beneficial if your practice work average is greater than your assessment average.
What does this mean?

• Grading practice work has the potential to be the most detrimental to students who show the strongest understanding on assessments.
• Grading practice work has the potential to be the most beneficial to students who work hard and do their job, but struggle to show their understanding on assessments.

I'm not at all interested in the first point; in fact, I'm not at all interested in using grades as punishment. I'm fine with the second point.
Which leads us to the new policy: I'll keep track of practice work, because it's formative assessment. I need to be able to see if students are learning the skills they need to "play the game" of performing on assessments. However, I'll only count it if it helps your final grade.

Reflection
I had students go back and check out their first quarter grades, and whether/how practice work affected them. This allowed me to address some really wild misconceptions ("It helped my grade went up by about 30%. It is very beneficial for me."), but also allowed most students to see that it has a very minimal effect, and that effect is sometimes negative.

For my own reflection, I had very mixed emotions after going through this with all 5 classes. A couple of them got it, and it felt like we came to a really powerful understanding. For others, including the class that inspired the problem, it was a big "Meh..."

If you've ever taught the same lesson 5 times in a row, you've probably had the experience of waning excitement on your own part: what seems fresh and exciting for the first two classes can start to seem repetitive and dull for the last two. I think that's some of what I was feeling, but there's also this vast difference in the dynamics of my classrooms that I've been struggling with all year. 
I hate that I devoted this much time to talking about grades, when I really need to get the focus of these students away from grades and onto understanding.

I love the problem itself, and the fact that it was directly inspired by a need expressed by the students. I'd like to try to incorporate more problems like this in my classroom instruction. Problems where students have to build a model of a real-world situation, work through the mathematics of that model, and use their results to come to a real-world conclusion.

I started writing the first part of these posts directly after teaching it to my first two classes, hence the title. I remember feeling like "Wow! Look at this amazing thing that's happening!" The third class later that day just totally burst that bubble (the class I wrote the problem for, the source of the most emphatic "Meh"s), and really brought me down. I didn't finish the post; within a couple weeks I had stopped posting on my photo blog and gone twitter-dark, and the rest of the semester was really an unsatisfactory struggle.

With some time to re-charge over the winter break, and re-inspire at a recent Math Summit in Shanghai, I feel like I'm ready to get back into things now. There's an exciting curriculum switch coming up at our school, which I get to play a big role in. I've got some engaging project/assessments going in all of my classes to give everyone a good start on the second semester. We just bought a bunch of potted plants for our house, which we're giving silly names. Things are looking up...