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Insights Into Algebra 1: Teaching for Learning

Mathematical Modeling Teaching Strategies: Listening to Students

According to veteran teacher Steve Reinhart, students can sometimes be the best instructors. In an April 2000 article in Mathematics Teaching in the Middle Schooltitled “Never Say Anything a Kid Can Say,” he wrote the following:

When I was in front of the class demonstrating and explaining, I was learning a great deal, but many of my students were not! Eventually, I concluded that if my students were to ever really learn mathematics, they would have to do the explaining, and I, the listening.

Listening to students, and using their responses to guide instruction, is a difficult, yet necessary, skill to acquire. It is impossible to promote learning without hearing what students have to say. The National Council of Teachers of Mathematics (NCTM) promotes active classrooms in which teachers gather information about students through questioning.

To ensure deep, high-quality learning for all students, assessment and instruction must be integrated so that assessment becomes a routine part of the ongoing classroom activity … In addition to formal assessments, such as tests and quizzes, teachers should be continually gathering information about their students’ progress through informal means, such as asking questions during the course of a lesson, conducting interviews with individual students, and giving writing prompts (Principles and Standards for School Mathematics, NCTM, 2000: p. 22)

Listening to student responses in class and during interviews helps teachers make informed instructional decisions based on what the students know and do not know. Listening also enables you to formulate questions that can help students develop deeper understanding.

Read what Orlando Pajon says about listening as a tool for assessment:

Transcript from Orlando Pajon

The purpose of the presentation, first of all, is that we want the students to tell us what they found out or what they learned. We don’t want the teacher to tell them what they need to learn. But for us, it’s also feedback. If we hear the student saying something different from what we expect, then we have to go back and either paraphrase or make sure that the student is understanding.

Effective Questioning and Listening

In order to use student responses to inform instruction, you must establish an environment in which students feel safe to respond. Ask questions that will encourage students to answer. Then, listen carefully to the student discourse so that you can frame additional questions that will bring a deeper understanding of the concepts and allow students to correct any misconceptions that they may have.

The following list of reminders for listening well and asking good questions is paraphrased from Steve Reinhart’s “Never Say Anything a Kid Can Say.”

Never say anything a kid can say!

This one goal will help to keep you focused. It requires you to develop and improve your questioning skills.

Read what Carol Malloy has to say about how Sarah Wallick uses student responses to guide a lesson:

Transcript from Carol Malloy

The [transmission factor lesson] was brilliant. Instead of just asking the students questions and having them give answers, or standing up there and lecturing about the entire content, she would lead them for a bit, and then she would allow them to work, and then she would question them on what they had found and use the information from the questioning to push the lesson to the next point. I was really excited about the way that she moved through every point that she wanted to get in a rather difficult lesson.

Ask good questions.
Good questions require more than recalling a fact or reproducing a skill. A student should learn by answering your questions, and you should learn about your students’ level of understanding based on their responses.

Listen to what Carol Malloy has to say about Sarah Wallick’s questions:

Listen to audio clip of teacher
Sarah Wallick

Transcript from Carol Malloy

Sarah has a talent for asking students questions and allowing them to say exactly what they mean, but also explain why they feel that particular way. There were several places in this lesson where students had to explain why they presented material the way that they did … When they talked about what they learned, they were very clear, and they had different things that they learned, and they all understood that they had to think about personally what this lesson meant to them and what was new for them … Generally, Sarah questions well. She doesn’t ask trivial questions. She asks questions that require analysis… she questions in a way that all of the major points of the lesson are given by the students. She does not deliver the major points. She asks them to deliver that to their peers, and they do. The way Sarah structures her lesson, she has to listen to her students, because her lesson depends upon what the students are saying. She has to be able to build upon their responses to the questions that she has, and questions that other students have, so that she can put forward the lesson.

Use more process than product questions.
Product questions – those that require short answers or a yes or no response, or those that rely almost completely on memory – provide little information about what a student knows. A process question asks the student to reflect, analyze, and explain her thinking.

Instead of giving a lecture, ask a series of questions.
Use student responses to disseminate the information that they need to know. If you think that a lecture might be a better format, ask yourself the humbling question, “How many of my students will actually be listening?”

Read what Sarah Wallick has to say about using questions to guide instruction:

Transcript from Sarah Wallick

One of the advantages of using this exploration model rather than direct instruction is that it allows students to be able to develop the ideas on their own. You remember the ideas that you come up with on your own a lot more than you remember things that other people tell you. And so I try, when we’re involved in a questioning phase or a discussion phase, not to ask really leading questions, and I try not to direct their answer. If I’m concerned about what they’ve said, I’ll direct that question to the students to address rather than making the correction myself, because that way the students are engaged in, “What is the issue here?” I already know what the issue is. My job really is to be a mediator in that discussion.

Read what David C. Webb has to say about Orlando Pajon’s use of questions:

Transcript from David C. Webb

You’ll notice that Orlando continues to draw from students’ informal ideas. And he also is allowing students to move forward and to give the full range of representations, from informal to pre-formal to formal. And he runs with it, he lets students show the connections between these different representations. And knowing that this is the capstone lesson, Orlando continues to be selective in what’s represented at the board and what he chooses to discuss. And even though there might be a range of insights offered in the student groups, Orlando continues to harvest the particular ideas that need to be emphasized in this unit.

Be patient.
Wait time is important. Most students need to time to process their thoughts and to gain the courage to raise their hands. By always calling on those who raise their hands first, you cheat those students who need more time to generate a response to your questions. When you show students that you are willing to wait and let them think about a question before they respond, you will be rewarded with more thoughtful responses.


In addition to the suggestions given above, list at least two other strategies you use to ensure that you ask effective questions during a lesson.

The Classroom Environment

A prerequisite to listening to students is encouraging them to share their thoughts. A supportive classroom environment, in which students believe that their opinions are valued, develops over time. Interaction will likely not happen on the first day of the school year, but with careful planning, students will grow more comfortable sharing their mathematical thoughts.

A corollary to the “be patient” rule is “be uncompromising.” While it is important to give students enough time to think about an answer, it is equally important to require students to express their thoughts. Allowing students to remain quiet during classroom discussions sends the message that their thoughts are not important and, more generally, that classroom interactions are not necessary for learning.

Listen to what Sarah Wallick has to say about promoting classroom interaction:

Listen to audio clip of teacher
Sarah Wallick

Transcript from Sarah Wallick

At the beginning of the year, my students were not as forthcoming … If I were to ask a question at the beginning of the year and say, “Well, what do you think about that?” or “Do you have more ideas?” everyone would sit with their hands folded on their desk and they’d look at me expectantly waiting for some sort of wisdom to come out of my mouth. And what I wanted was wisdom to come out of their mouths. I had to be really aggressive, and if no one would raise their hand, I would just go to them and say, “What do you think?” And if they looked at me and said, “Nothing,” I would look at them and say, “No. No, you have some sort of thought – are you thinking about what day it is?” And then they would finally confess that maybe they were having a mathematical thought. And when they confessed that they’d had a mathematical thought, then we had something to launch with. In a new classroom situation, they don’t know each other, they haven’t built those safety relationships, and so you have to get them to build them. The only way to get them to build them is to have them say something and find out that it’s safe to do that.

Once students begin to express themselves in the classroom, they will continue to do so only if they can see that the teacher values their responses. It is a teacher’s responsibility to provide an environment in which students feel safe to share ideas.

In addition, students must understand that everyone in the classroom has a responsibility to listen carefully to every speaker. For successful interactions, students must listen to the teacher, and students need to listen to one another, too. Often, in the course of discussion with a classmate, a student will recognize an error in their thinking. And many times, an explanation from a peer will be far more effective than a lecture from a teacher.

Read what Sarah Wallick has to say about students listening to one another:

Transcript from Sarah Wallick

When a student comes out with a really good point and nobody else is listening except me, then nobody else learns anything. I work hard just to get kids to listen to each other, so I rarely repeat what a student will say. If I think a student isn’t listening to what a classmate is saying, I’ll look at the person who is not listening and say, “What did you think about what he said?” I’ll just put them on the spot. I do this a lot in the early part of the year particularly, because I want them to understand that it’s really important for them to listen to one another.

Read what Carol Malloy has to say about students speaking with one another about mathematics:

Transcript from Carol Malloy

There was a conversation going on about, “Well, how do you know how many times this [pulley] goes around? How do you figure it out in the very beginning of the problem?” And the young lady said, “Well, you simply have to use the circumference of the circle in order to figure that out.” And they did that. You had students who actually talked to one another about the mathematics in the groups and across the entire class. And there’s nothing more that you can ask from a community of learners and everyone learning together, listening and respecting what everybody else is saying, and valuing the fact that somebody else has something to contribute to their knowledge and their growth.

To promote classroom interaction and foster a safe environment, teacher and peer reactions to a student response should be nonjudgmental. Most teachers understand that negative reactions are hurtful – an unsatisfactory answer from a student does not require a harsh critique, especially in front of the rest of the class. But it is important to remember that a positive reaction can also have a deleterious effect on class participation. Imagine being a student in a math class, and the teacher says to a classmate, “Wow! What a fantastic answer! That’s tremendous! Okay, who’s next?” How likely would you be to raise your hand and attempt to follow that? Even the most confident student wouldn’t want to have to go next, let alone the shy teenager who lacks self-confidence.

Read what David C. Webb has to say about developing an effective classroom environment:

Transcript from David C. Webb

Such a classroom is not developed overnight, and students don’t necessarily come into a teacher’s classroom ready to engage in these types of lessons. It’s developing an atmosphere of student inquiry in the classroom where students are allowed to pose ideas – whether they are correct or not, it doesn’t matter. “Give me your thoughts on this.” “What do you see?” These are questions that all students can engage in. Students will give feedback to students and eventually, in the whole class discussion, there will also be opportunities to address some of those misconceptions. It’s a way of engaging in the learning of mathematics that’s safe for a wider range of students. It creates mathematics that’s more accessible to students.


Explain how you might implement one of the strategies discussed above to increase the potential for learning in your classroom.

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Insights Into Algebra 1: Teaching for Learning


Produced by Thirteen/WNET. 2004.
  • ISBN: 1-57680-740-1