From what I understand, years 3-5 can be among the most transformative years in a teacher's career. That lands me right in the middle of that range. And despite my distaste for the New Mexico teacher evaluation system, I am thankful to my current principal, who uses it effectively as a way to reflect upon and discuss my practice. Recently, during my post-observation meeting, I sat down with my principal and we decided the next areas for me to focus are tweaking my questioning techniques and taking a deep look at methods for assessment.

With that said, I have been really focusing on low-barrier versus high-barrier questions when engaging students in a lesson. I found, that in my first two years of teaching (starting to breakaway in year 3), that I was so ready to introduce students to new content that I would introduce new procedures before I even gave students a chance to use their own reasoning and demonstrate what they already knew. I was so caught up in the idea that a diagnostic had to be a written test and I didn't want to waste a class period so why not quickly move ahead?

Now, in my fourth year, I have really made it a focus to open up new ideas by giving students an opportunity to try their own strategies for solving problems I will ultimately give them a new strategy for solving. Recently, the idea I wanted to introduce was using a line of best fit to make predictions about future data. Old me would have simply said today we are going to learn about lines of best fit and then I would move full speed ahead, using the "I do, you do, we do" method of instruction" (I shudder just thinking about my confidence in this strategy as the only logical form of instruction). Now, I present my students will the following problem:

The Google Doc version of this file can be found here.

The goal is for students to come up with their own strategy for predicting these future data points and some of the ideas are incredibly sophisticated and show a deep level of understanding. Some students said they noticed a positive correlation and estimated where the points would be at 750 and 1400 minutes. Others created an elaborate box structure to create a high and low option for each set of minutes and then put their point in the center (if I remember, I'll post a picture of this). and still others, estimated by drawing a line through the top and bottom points to estimate future values, showing they were already thinking about lines of best fit.

Year 3 me would have said, "Cool! Thanks for working through this guys (without having students show off their work). Now let me show you the correct way to do this!" Today, wiser year 4 me takes mental notes of different lines of reasoning that students came up with on their own, which they then share with the class. In this way, I am not the owner of new knowledge. Student ideas are validated and multiple lines of access to the content are opened up. Now that students have had a chance to develop their own reasoning and share it with the class, they are primed for me to demonstrate the alternative method of using a line of best fit (which some students came incredibly close to generating on their own).

In the end, opening with students presenting what they think is a powerful motivator and leads to deeper initial understanding of the underlying concepts in new material. This is similar to my opening activity for linear equations seen in an earlier post about constant difference. I find that even my lowest level learners are capable of accessing mathematics when they aren't required to "crunch numbers" right at the beginning, but can reason through the process. I am going to continue to explore these opening activities and try to generate even more. Maybe this will become a serious hobby and lead to a blog in the style of Visual Patterns produced my Fawn Nguyen.

I just want to give the MTBoS a shoutout for pushing me to be better and move beyond teaching how I was taught.