I had what I think was my best moment of teaching this year yesterday.
Scene: Day four of scatter plots. Day one we played with gapminder data. Day two they took quizzes to wrap up univariate data and talked about graphing bivariate data and an intro to association/correlation. Day three we talked about positive/negative/no association, and linear/non-linear association, with some discussion of how these relate to best fit lines.
Big, concrete goal of day four is that students can identify and describe outliers. Once I felt good about that we would move on to graphing bivariate data to draw conclusions about it in preparation for a deep dive on lines of best fit next week.
I gave kids the page below — it’s a question cut from the Ready Common Core books, with my prompt to explain using a sentence starter after. I started using sentence starters like this one after reading this post by Doug Lemov–it’s made a huge difference in my student’s ability to articulate mathematical reasoning.
My standard practice when I use this technique is to give kids a minute or two on their own to answer the question and write their sentence. I circulated, found a few sentences I liked, and asked their writers to share out, then took volunteers who wanted to share their sentences as well. I wasn’t surprised that almost everyone picked the right answer. I was surprised about the quality of their thinking — several students shared sentences describing how point D was an outlier because it didn’t fit the pattern of the rest of the data, instead of simply saying it was far away from the other data points. I see this as the next level of reasoning around outliers — it’s not just that they’re isolated, it’s that they don’t fit with the association of the rest of the scatter plot.
So I decided to improvise. I threw up a scatter plot that looked something like this:
and asked if my students thought there was an outlier. I think about two-thirds of hands instantly shot into the air. I almost called on someone right away, but resisted the urge to take the cheap points and asked students to turn and talk to someone next to them — is there an outlier or not, and why. An immediate, unanimous “no” went around the room. I gave them a moment to explain their reasoning to their partner and brought the class back together. Now almost every hand was in the air. A number of students explained why the point was not an outlier, and showed strong reasoning connecting it to the meaning of association and predictions based on the data. I would normally summarize key points here and give some notes to feel like all the students who didn’t participate got the key ideas, but I skipped it. I told them how impressed I was, and moved into some more advanced practice graphing and interpreting sets of bivariate data.
Maybe this doesn’t seem groundbreaking. It isn’t the most important mathematical insight they will have of the year, or close to it. And plenty of students have light-bulb moments like that all the time. I think there are a few big reasons this class blew me away:
1. It was everyone. And I mean everyone. I was trying hard to find someone who was confused but I couldn’t, even when I stretched it with the more complex outlier question.
2. Students were using language that basically pre-taught key concepts from our next topic — lines of best fit. If they understand the power of a scatter plot to predict data points we haven’t observed, they’ve made a huge step toward the goals of next week.
3. This class doesn’t love to participate, beyond about four kids. And a bunch of the kids with their hands shooting up have failing averages–despite hard work and perseverance on their part.
4. Finally, this reflects some of my deepest held beliefs about quality teaching. It’s not a mysterious art, it’s a serious of deliberate, planned but flexible teaching moves that set students up to think hard about challenging questions and construct flexible, powerful understanding.