Jennifer Wilson wrote a post about productive struggle that you can find here. I did my usual trick of sending myself the link. I have a filter on my school email that automatically labels emails from me and files them away under "Twitter" so that I never actually see these emails. Thankfully I keep vague recollections of what I have tagged and earlier this week I knew to search my "Twitter" label for something interesting relating to sine law.

I stole this from Jennifer's post, having first shown my students the picture of the riverbank and surveyor and the diagram without any numbers. Straight from Jennifer's post.

Students worked in groups on their big whiteboards to find the width of the river. There was definitely some productive struggle going on and a lot of good mistakes were made along the way. I love that students try to use other mathematics they know, in this case Pythagorean theorem and right angle trigonometry. There were some good conversations around when those apply and why they don't in this case (with the triangle as drawn here). One group split up the 107° angle into a 90° and and a 17° angle (below). They struggled a lot and did some good math along the way. When they realized that they needed to create an altitude instead, they were quick to solve the whole thing.

Groups, one by one, figured out that they needed to add the altitude from S2. Some made mistakes along the way...

But they eventually got there!

Once they had found the width of the river, they had to work with an oblique triangle that did not have any numbers on it (again stolen from Jennifer's post):

Find an expression that represents the length of side *c*, I said. I got a lot of, uh, interesting looks. Most did not want to work without numbers. Didn't know where to start. They struggled in their groups for a bit then I told them to finish it for homework. That was their only homework.

The next day, we worked through the development of the sine law together and it went fairly well - better than in previous years, I think.

Working in groups on the big whiteboards has really made a difference to the culture of my classroom. Students are willing to try things and make mistakes and they persevere. That makes me happy.

As a side note, none of my students thought of solving it the way Jennifer showed on her blog here. I will have to make sure I work that in next time around.
There have been a few tweets lately that have made me pause and think about tests. Even in the course where I spiral through the curriculum and no longer have units, I still give tests. I will admit that I'm not sure how to evaluate without using tests given the time constraints in place and the rigidity of the department in which I teach. I would love to sit down with each student and ask questions to elicit what they understand and where they are struggling, but I don't know how to manage that during class time. I look at Alex Overwijk in awe as I know he does this successfully.

The tests I give, even in my spiralled class, look fairly traditional, but the way I "administer" (that sounds so formal when, in fact, I spend the entire class running around from one student to another) is likely not. The students in this class, grade 10 applied, all get two classes to finish the test. I take the time pressure off for everyone. I also don't let anyone hand in a test without answering all the questions, which means that I prompt as needed. I write whatever I ask or say in a pink or orange pen on their test paper and take it into consideration when I am marking the test. This means that the students who say that they don't care or that they have had enough still have to do work. I don't accept them opting out, especially since I know that they all can be successful. (So why am I testing them?) I don't have the same flexibility in my other classes though...

(I own this shirt)

In my academic classes (MPM2D and MHF4U this semester) I like to do stations for review the day before a test. The questions come from last year's test. I know the old tests are out there, some students have them, others do not, so I feel that this helps level the playing field. They work on the questions in random pairs on big vertical whiteboards and show me only the answer (unless the question asks for a proof in which case they bring the whiteboard to me!). They get a sticker when their answer is correct. Stickers provide motivation in ways I still do not fully understand. Here's the thing - when they do not get the correct answer I try to give them some feedback that will help them find their error and fix it. For example, if they had to determine the sine equation given certain information, I would tell them that the amplitude and vertical translation were correct in their equation, but they needed to look at the phase shift and period again. If they came back and the period was still incorrect, I would ask more specific questions about the problem to help draw out what they know about period then have them work on the phase shift again. **This process of letting students correct their work in real time seems incredibly valuable to me.** So I wonder why we cannot do this during a test. I realize that the logistics of attempting to do this with 30 students in 75 minutes do sound challenging. But I take issue with the snapshot of learning that we get with tests. Have you had a student do poorly on a test despite knowing that they understood far more than they were able to show you on the test? I have. Over and over.

I have started caring less about what I am supposed to do and more about doing what I think is right. When students don't succeed the first time, I give them another opportunity to show me what they have learned. After the 2nd grade 10 academic math test this semester, one student in particular had trouble with several questions bringing this student's overall level on the test below where I thought it should be on this material. I had the student come in at lunch and we talked about how to tackle a couple of the questions. We had a good discussion - I found out what really was not understood and what this student could do given how to start the question. I think this was such a positive step forward for both of us. The student cleared up some misunderstandings and built up confidence by being able to show me all the things that they could do, but hadn't. I learned more about that student's understanding and what they really knew and showed that the learning, not the test, was the important part.

With what have other HS math teachers replaced tests? I would love to hear options. I teach at a school where the pacing guide is king, so I have an uphill battle, but I think this is the next goal in improving how I teach. I need help and would love to hear your ideas. Thanks in advance.