Category Archives: Mathematical Practices

4 ways to improve classroom conversations

Although I loved the first day of school, September was always a difficult month for me as a teacher.  It’s a month of establishing classroom norms–the rules and expectations of your classroom.   What it means to have a conversation about math in your class may be different than what it means in another class.

As you begin to negotiate the norms in your class regarding classroom conversations, I wanted to share some insight from the research about how to make norms you may already be familiar with even stronger.

Two researchers (Kazemi and Stipek, 2001) observed four classrooms that encouraged classroom conversation about math.  However, the quality of the math talk in the classes differed.  Some classes had norms that encouraged a “high press” for conceptual thinking, while others did not.

Here are four things they observed about how to promote conceptual thinking:

  1. Explanations:  While both types of classes asked students to explain their work, high-press classrooms set the norm that explanations needed to be mathematically based.  Repeating a rule or citing a textbook or prior teacher did not count as an explanation.
  2. Multiple Strategies: While both types of classes, encouraged students sharing strategies for solving a problem, high-press classrooms focused on what was mathematically similar and different between the strategies.
  3. Student Mistakes:  While both types of classes created a culture where it was ok to make mistakes, high-press classroom used mistakes as an opportunity for students to try out other strategies or explore contradictions. The teacher did not validate or invalidate the solutions, but presented both and then asked to students to determine which was correct and why.
  4. Collaboration:  While both types of classes encouraged students working together, high-press classes focused on both individual accountability (every student must be able to explain the group’s solutions) and coming to agreement as a group through mathematical argument (as opposed to voting on which they liked best).

As I mentioned previously, change takes time.    It will take a while for students to become comfortable talking about math in these ways.

What do you think?  What norms do you set for classroom conversations?

Read more here:  Kazemi, E., & Stipek, D. (2001). Promoting conceptual thinking in four upper-elementary mathematics classrooms. The Elementary School Journal, 59-80.

Making New (School) Year’s Resolutions Stick

I love the first day of school.  I love sharp new pencils and blank notebooks.  I love meeting new students that will change your life in ways you can’t even begin to imagine.   Most of all, I love the hope it presents.  It’s a chance to do things differently.

Many of you are excited to try new things this year that you learned about over the summer.  Maybe you attended a workshop.  Maybe you are using a new inquiry-based curriculum.  Maybe you took Jo Boaler’s course on “How to Learn Math.”  Maybe you have been reading the blogs.  Maybe you have been reading the research.

In any case, you are ready to start the school year with a more student-centered approach to math.   Teaching with less of an emphasis on memorizing procedures and more of a focus on developing an understanding of the math concepts.

Here’s the thing.  It’s really hard.  And so the year starts off with the best intentions and then they fall by the wayside come October.

It reminds me of my New Year’s resolution to work out more.  Last January, I took a kickboxing class.   The first thing the instructor asked us to do was ten push-ups.  The problem was I couldn’t do one push up. Not one.  While everyone around me did perfect push-up after perfect push-up, I lied on the mat completely frustrated.

So what did my trainer do?   Did he tell me to give up and go home?  Of course not. He told me to try a push-up on my knees.   That first day, I could barely do three. Then the next week I could do more.   A few weeks later, he had me get off my knees and do a push-up where my legs were far apart.  Each week, I moved my legs closer together until one day I finally did one push-up in perfect form. It took months.

I share this because real change takes time.    There is no quick fix when it comes to teaching and learning math.  

Many students aren’t used to a math class that involves problem solving.  They aren’t used to the idea that struggling with tasks can be a good thing.    They aren’t used to looking for multiple solutions.  They aren’t used to justifying their answers.  They aren’t used to having classroom conversations about why a procedure works or whether it works all the time.

If you try to completely change the way students do math all at once and expect change to happen overnight, the kids are going to lie on the mat frustrated.

Just like I needed to build the muscles in my arms and core, your students need to build the muscles in their brains when math class no longer involves the teacher showing them what to do to and helping them as soon as they get stuck.

Building this new culture takes time.  Developing new norms in your classroom will be an on-going process.  Don’t try to do everything at once.  Students may not know how to approach a problem when they aren’t told what to do.  They may not know how to talk about math or what an explanation is or how to justify their answer.

That’s ok.  You will get them there as long if you don’t give up.  Start by adapting problems so they have multiple entry points. Give them smaller tasks at first that will build their stamina.  Then move to more complex problems.  Start with little bits of math talk and have the conversations get a little bit longer each week.  Most importantly, give them time to adjust to a new way to learn and do math.  One day, they will be able to do a push-up in perfect form, but it’s not going to be right away.

I’ll talk more about the specifics of how to begin to do this in another post, but if you want to know now, check out this post and research below.

Carpenter, T. P., & Lehrer, R. (1999). Teaching and learning mathematics with understanding. Mathematics classrooms that promote understanding, 19-32.

Wood, T., Cobb, P., & Yackel, E. (1991). Change in teaching mathematics: A case study. American Educational Research Journal, 28(3), 587-616.

Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for research in mathematics education, 458-477.

 

How do you know? Part II

In a previous post, I talked about why it might be important to ask a student “How do you know?” when they give an answer to a problem.  What I neglected to talk about was the research that supports this recommendation.

Luckily, Education Week recently published an article about several research studies that claim asking students for explanations can deepen their understanding.

Although the research mentioned does not talk specifically about learning math, it does lend support to the idea that asking students for explanations is important.  Furthermore, one study suggests that asking for an explanation focuses a student’s attention on what the underlying concept is as opposed to other aspects of the task.

What I did not see addressed in the research presented was how we can better foster explanations in our students.   It cannot be enough to simply ask them for explanations or to focus their attention on figuring out how something works.  It seems to me that we need carefully designed tasks that allow them to develop an understanding of why something works so that they can explain it.

Asking for an explanation is a good suggestion, but what do we do when students can’t explain?  I’d love to hear your thoughts.

Want to know more?

Access the article (which includes links to the accompanying research papers) here.

How do you know? Making thinking explicit

When you watch a typical math class, you see a lot of rapid fire questioning going on.
Teacher: Number 1?
Student: Five and ¾
Teacher: Number 2?
Student: 3 and ¼
Teacher: Number 3?
Student: 2 and ½
Teacher: Anybody get a different answer?
Student: 2 and ¼ .
Teacher: Good. Next problem?

Now in some classrooms, a teacher will ask a student to come up to the board and show their work. Sometimes a student will even be asked to explain their work. However, that often sounds like this: “First, I looked at the denominators. They weren’t the same so I made them the same by multiplying. Then I added the numerators but left the denominators the same.”

This explanation is a summary of what the student did, but it doesn’t give any insight into how the student thought about the problem. Why did the student make any of these decisions along the process? How did he know why to change the denominators and not the numerators?

For the students who didn’t get the correct answer, this explanation doesn’t help. They just heard a list of steps that they need to memorize and remember correctly next time. The process of why these steps work is a mystery. It’s a black box.

For the students who can explain why it works, they are often not given the opportunity to explain. They are never asked to make their thinking explicit. What if they were given opportunities to try to explain their thinking so that it becomes an explicit process instead of a black box? Would all the students in the class benefit from it? I think they would.

At first this process can be uncomfortable. Students aren’t used to talking about their thinking. It seems private, something that no one asks you about.

But slowly, if we ask questions like:
Why did you do that?
• How do you know that is the correct answer?

They will begin to be comfortable talking about their thinking.

Now if the students only know because they memorized a set of steps, that is all they will be able to explain. But if they are given tools to start thinking about why things work, asking them how they know can change the way a classroom operates.

If the goal in math is to figure out how rules work or why they always work, the questions in math class change, the activities change and everyone is engaged in trying to discover what is in the black box.

I’d love to hear what happens when you ask students, “How do you know?”

Is this answer right?

“Ms. Placa. Is this right?” I must have heard that sentence a million times when I began teaching.

Many classrooms create a culture where the teacher has all the answers and the students seek approval from him or her. This sometimes works out well for things like learning about conventions or notations that the student has no way of knowing.

But it doesn’t work for learning what it means to do math. It creates a misconception among students that math is a set of arbitrary rules that the textbook or teacher tells them. The goal of math class becomes to figure out what the teacher wants you to do.

Children pick up on how to play this game early on. They note the teacher’s body language and tone of voice when the teacher responds to an answer. For example, in many classes, when a teacher asks, “How did you get that answer?” it is a signal that you should change your answer. You must have made a mistake. If you didn’t, the teacher would have just went on to the next problem.

What if we taught students that the goal was not to please the teacher, but rather to convince themselves and others why their answer was correct? The rules of the game would change. It would be the teacher’s role to give you tools and tasks that fostered learning but students would be responsible for determining if they solved a task correctly.

Yackel and Cobb (1996) give a great example of how to begin to do this. A student gave an answer to a question and then wavered when the teacher questioned her. She understood the question as a social cue to change her answer. Below is the conversation they had after she changed her answer:

Teacher: Wait, listen, listen. What did Mr. K.-what have I always taught
you? What’s your name?
Donna: My name is Donna Walters.
Teacher: What’s your name?
Donna: My name is Donna Walters.
Teacher: If I were to ask you, “What’s your name?” again, would you tell
me your name is Mary?
Donna: No.
Teacher: Why wouldn’t you?
Donna: Because my name is not Mary.
Teacher: And you know your name is—…If you’re not for sure you might
have said your name is Mary. But you said Donna every time I
asked you because what? You what? You know your name is what?
Donna: Donna.
Teacher: Donna. I can’t make you say your name is Mary. So you should
have said,”Mr. K. Six. And I can prove it to you.” (p. 468-9)

This is a great example of how to begin to create norms in a classroom so that students’ explanations become the focus of classroom discussions. Convincing themselves and others that their answer is correct becomes the role of the students in the class. They do not need to rely on the teacher for approval.

How might this change the way students think about doing math? How might it change the way we think about teaching math?

References

Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for research in mathematics education, 458-477.