The demands on teachers have increased exponentially in the last decade. Teachers are expected to teach at high levels of cognitive demand to help all students meet new rigorous standards. Elementary teachers are expected to be experts in multiple subjects. This is particularly true in mathematics, in which teachers may not have strong backgrounds. In most states, only one math course is required. Meanwhile, new professional standards require teachers to organize classrooms in ways that promote peer-to-peer discourse and reasoning in math, as well as to address the variety of individual student needs.
To help teachers improve their math teaching, a great deal of money is distributed to provide professional development. Last year, it is estimated that $18 billion was spent on professional development (PD) in the U.S. And while much of it was excellent, studies such asquestion the extent to which PD actually changes behavior in the classroom or student achievement. Even the best professional development can't fully address the daily needs of the teacher. What is needed is a curriculum that includes strong questions for students in daily lessons.
Asking Better Questions
One of the ways we can support teachers is to help them ask better questions that stimulate discussion and mathematical reasoning. Too often, the questions we pose have a single answer or require simple memorized facts. How can we ask questions that support conceptual understanding and make sense of the mathematics? There are many lists of such questions—generic questions such as “Why do you think that?” or “Is there another way of explaining that?” and so on. However, it is also critical that questions address the specific mathematical concept and content. This requires a strong background knowledge of mathematics and the specific questions that can enable students to make conjectures and generalizations.
Let’s look at an example. In the past, when teaching rounding in the third grade, we would just provide students with the following rule: Look at the digit to the right of the one we are rounding to, and if it is 5 or higher, round up. The question we would ask is, “What is 353 rounded to the nearest hundred?” In other words, it was a procedure that merely required students to apply a rule without any conceptual understanding or reasoning. A question at a higher cognitive level would be, “What are all the numbers that round to 400?” (assuming this is a third-grade class learning about whole numbers).
Before we ask that, though, what do we mean by rounding? Really, all we are asking is which ten or hundred or thousand is the number closest to. What if we asked students to use a number line where they write numbers that are between 300 and 400—two closer to 300 and two closer to 400—and then asked them how they knew those numbers were closer? What if we asked them to prove that 351 is closer to 400, and 349 closer to 300? What if we asked how students knew a number is closer to 300? What if we asked students whether 350 is closer to 400 or 300? Wouldn't that be a lively discussion, given that 350 is the same distance from 300 as 400? The next question might be, “What should we do when a number is right in the middle, not necessarily closer to either rounded number?”
With the right content questions like these, students don’t just learn a rule—they develop an understanding of why the rule works. Students can use a number line to recognize that all the numbers between 351 and 449 are closer to 400 than to 300 or 500. Only 350 and 450 are the same distance to 300 or 500, respectively, as 400. Therefore, you cannot say 350 is closer to 400 than it is to 300. It is numbers right in the middle that require a common agreement. At least in math class, we will round up when the number is equidistant from two numbers.
It is also clear that rounding up is arbitrary and not always done in the real world. Certainly, the bank and grocery store don’t round up. It would be nice if these questions were presented in a teacher’s guide, the most important question being, “Which number is exactly in the middle between 300 and 400, and how should we round it?” As students work with a partner, the teacher walks around the room prepared to scaffold and stimulate conversation between students.
3 Types of ‘Good’ Questions
Good questions in the math classroom might be divided into three types:
- Literal questions ask what students are doing and thinking: “How did you know 223 is closer to 200 than 300?”
- There are also questions that support student thinking, or scaffolding questions: “Which number is closer to 223: 200 or 300?”
- Finally, we have questions that ask students to hypothesize or generalize: “Let’s look at a number that is in the middle and determine the closest hundred.” This should stimulate a conversation about how 250 is not closer to 200 or 300, leading students to come to a common agreement.
Teachers are familiar with asking questions in a whole-group setting, but less so when students are working with a partner or group. In the latter case, the teacher must decide if these students need more scaffolding or if they need to more clearly share their understanding. Which questions will best develop understanding and support discourse?
Posing real questions, such as what should we do when a number is right in the middle between two numbers, enables students to reason, defend an argument, and critique the reasoning of others. Students become mathematicians. Helping teachers ask more open-ended questions—questions that don't merely require recall or memorization—is one of the goals of an effective teacher’s guide. Students are asked to defend an answer, explain how they arrived at an answer, and consider several different methods.
Embedded questions, like the above examples, help teachers promote mathematical understanding and discourse in the classroom. Students work together and look to each other as well as the teacher for information. Good questions ask students to reason, to conjecture, and to consider different strategies to solve problems.
Eventually the goal is for students themselves to ask their own questions. In the words of George Polya, the father of problem solving, “But I think there is one point that is even more important: Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics.”
Interested in learning more about Singapore Math®, which encourages a curriculum that promotes student discourse?for our webinar, “Supporting Teachers to Create Classrooms That Promote Student Discourse, Deeper Understanding of Math Concepts, and Problem Solving,” presented by Andy Clark and Christopher Coyne on April 16 at 4 pm ET.