2013 NCTM Conference (4/19/13 - 12:30 pm - Mark Howell):
I've come to believe that asking the right questions is one of the most important parts of building student understanding of new mathematical concepts. One of my goals in coming to the NCTM conference was wanting to continue to sharpen my questioning skills. I was therefore eager to learn from Mark's session about "questioning the questions".
Mark started the session with a bold statement: he suggested that as beneficial as multiple representations of mathematical concepts are, the use of multiple representations is not sufficient. He argued that linking between and among the representations is where learning takes place, when student understand how the representations are connected together.
Mark proposed that good questions are those that come from powerful connections between representations. These questions prompt students to explore these connections and examine what the representations actually mean.
Mark noted that it is possible to ask a lot of questions in the classroom without building a deeper understanding of mathematics. He noted that we've all seen procedural questions that ask students to provide the next step in a familiar mathematical routine. These questions tend to be very repetitive and removed from context. But students can memorize procedures without knowing how to think about math or understanding what the procedures really mean.
Teachers often provide shortcuts for students that allow them to solve problems without thinking about how math really works. This can be detrimental for students in the long run. For example, students may have memorized the "vertical line test" without knowing the properties of a function.
Mark noted that questions are essential for revealing deeper mathematical realities. But it is difficult to craft good questions that prod students into these deeper realms. There are four different levels of questions:
1. Questions that elicit feedback from students.
2. Questions that ask for factual responses.
3. Questions that ask students to explain a procedure.
4: Questions that ask a student to explore math, make conjectures, explain and verify their results, or talk about connections between representations.
Most teachers' questions remain in the first three levels, but these questions are all very shallow compared to the fourth level of questions. It is important that we consider how to craft questions that take students to this deeper level.