Anyone who’s ever sat in a classroom or been a teacher knows that teachers ask lots of questions. And most of the questions that they ask are fact, recall, or knowledge questions that require very little thinking and reasoning. But the Common Core State Standards in Mathematics call for teacher questions and responses to student questions that prompt mathematical thinking. So what exactly does that look like?
There’s been quite a bit of research about the questions teachers ask, but Meredith Gall (1984) reminds us that answering a question is a 5-step process (listed below)that must be worked through.
- Listen to the question.
- Understand what is being asked.
- Answer to self.
- Answer out loud.
- Rethink and revise the answer for questions that are at higher cognitive levels.
As Gall’s list implies, effective questioning takes time. And probably the most useful thing that can be done to assist thinking is to do nothing but intentionally and systematically provide ample time for students to progress through the process and think about the question and their response.
In terms of the questions asked, let’s first look at the questions students ask teachers. I’ve found that most teachers, like me, tend to overlook the relevance and importance of these questions. But the way a teacher responds to a student question can be pivotal in encouraging mathematical thinking, independence, and instilling a confidence in them that they can do mathematics.
For example, if a student has performed an incorrect computation and asks me if the answer is correct there are myriad of possible responses I could provide. Note the difference in the responses in Figure 1 and Figure 2. As you’ll see, Figure 1 responses provide very little encouragement for students to employ mathematical thinking and reasoning whereas Figure 2 provides thought provoking questions about the problem. The responses to the questions in Figure 2 can be telling in terms of what the student does and does not understand about the process or concept.
Carefully posed questions can assist students in conjecturing and communicating their thinking and reasoning about the work they’ve done. I’ve found NCTM’s Professional Development Guidebook for Perspective on Teaching Mathematics (2004) recommended three generic response questions that teachers can ask at varying grade levels about multiple mathematics topics to be very powerful in fostering mathematical thinking. They include:
- How did you work that out?
- Could you solve that another way?
- Could you generalize your result?
Another piece of the questioning puzzle is the way a teacher frames or sets up questions they ask students. Consider the two questions shown below.
Both questions require the same mathematics for determining mean. But Question A focuses on the process of finding mean and is a closed question that has only one solution. A student could answer this question by simply applying the procedure for finding mean and really know very little about what mean is. Clearly Question B requires more mathematical thinking and reasoning in deciding which numbers would give a mean of 12, and is more open in giving students the option of using a variety of numbers, including those other than whole numbers.
Some other questions that align well to the Common Core State Standards in Mathematics and would serve us well today in developing mathematical thinking were developed by Watson and Mason in 1998. They include the following question stems that can be used with a variety of mathematics topics and grade levels.
- Show me or give me an example of …
- What other information is needed in order to answer the question?
- What is the same and what is different about…?
- Is it always true, sometimes true, or never true that…?
- Sort or organize these according to …
- Why do …, …, and … all give the same answer?
Quality questioning takes time and effort. But it’s time well spent not just in developing mathematical thinking, but overall problem solving skills that will enable students to continue to learn throughout their lifetime, so it’s well worth the investment.