The Questions I Still Ask…and Shouldn’t!

Several weeks ago in my statistics class at different points in the lesson I asked the questions, “Does everybody understand?” and “Does this make sense?” After each question, multiple students responded with clear explanations articulating the important ideas from the lesson. And then I awoke from my dream because this NEVER happens after I ask those questions! Almost always I gain very little information about students’ reasoning from asking these questions. I recognize this is a problem and I am working to find solutions. I have told my students that if I ask the class either of those questions you are more than welcome to ask me, “What is it that you want me to understand?” or “What is it that you want me to make sense of?” The greater challenge for me is that I need to think about better, more specific questions, that help me get a sense of what I want my students to truly understand. The charge I have given myself is to think about the three to five most important understandings I want students to gain from the lesson and create questions that will help me gain information about their understanding of those ideas.

This charge has forced me to think more deeply about what I want students to understand and how I need to structure learning to make this happen. For example, in statistics, this past week we spent time examining the meaning of correlation, a measurement of the strength and direction of a linear relationship between two variables. At the end of the explanation instead of  asking, “Does everybody understand?” I asked “What would be the correlation between two variables whose scatterplot revealed a relationship represented by a horizontal line?” Instead of simply a head nod response to “Do you understand?” , the question forced students to reflect deeply on the meaning of the measurement of correlation and the meaning of strength, direction, and linear relationship in this context. The ensuing discussion required students to present an explanation that provided clear evidence of their understanding. I would consider this a success because I learned so much more about their understanding of correlation and I believe the ensuing discussion deepened their understanding of the idea. Also, it forced me to think more deeply about what I wanted them to understand about correlation.

– Patrick

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Teaching Badly

“The depressing thing about arithmetic badly taught is that it destroys a child’s intellect and, to some extent, his integrity. Before they are taught arithmetic, children will not give their assent to utter nonsense; afterwards they will.”  —  W.W. Sawyer, Mathematician’s Delight

This quote makes me think of those milestones in a child’s mathematical education when she gets several unintended truths about math. One is that math is about following procedures and memorization. Another is that speed and efficiency are valued above all. Another is that math needn’t make sense. Maybe the greatest is that not everyone is good at math. This quote also reminds me of Christopher Danielson’s post on standard algorithms.

One of the most concerning problems is a collective agreement that, at least at the lower grades, mathematics is arithmetic. I would venture to claim that, based on my limited experiences, the general public conflates mathematics and arithmetic. My wife does or at least she enjoys pointing out how bad I regularly am at arithmetic, despite my mathematical education. I think many or most elementary teachers do, too. How do we change this for teachers for whom the bulk of their experiences as math students is/was in classrooms where arithmetic was the goal of mathematics when our opportunities to change the beliefs of these very important and influential teachers are limited to one or two semesters?

None of these ideas are revolutionary. We thought this might be a good place to start. Ease you in to all the revolutionary ideas to come.

– Adam