Monthly Archives: May 2013

Be a student this summer: Take a free online course on learning math

Looking for some free professional development this summer?  Jo Boaler, a professor of math education at Stanford University, is offering an online course for teachers and parents titled “How to Learn Math.”

In an earlier post, I mentioned that I heard her speak at a conference this year.  I admire the fact that she’s a researcher who tries bridge the gap between research and practice.  The summer course sounds like an interesting example of how she continues to try to make that connection.

Over the course of 8 sessions (lasting 10-15 minutes each), she plans to discuss some of the current research and explore best practices.   It’s self-paced and there will be opportunities to collaborate on discussion forums.

I wanted to share this with you because I’m often asked where to go to learn more about teaching and learning math.    Unfortunately, I don’t often have many quality suggestions.   Although I can’t vouch for this course because I haven’t taken it, I wanted to share it because it sounds promising.  In addition, it’s free and can be done remotely so there’s not much risk involved.

I recently signed up for it and I’ll be sure to report back, but it’d be more fun if some of you took it too so we could discuss.

Want to know more?

Click here to enroll for the course

Click here to learn more about Jo Boaler


Does everyone understand?

How many times have you asked your students that during a lesson?  I know I asked my students that question too many times to count.  But I never thought about what I meant by the word understand.

Richard Skemp talked about two different types of understanding back in 1976.   The distinction is still important today.  The first type is relational understanding: knowing what to do and why.  The other is instrumental understanding: knowing a rule and how to use it but not understanding why the rule works.

You can probably think of many examples of instrumental understanding in a math class.  Students may know to “flip and multiply” when dividing fractions and may be proficient at computing the answers but they have no idea why the rule works.

Why might it be problematic to only develop instrumental understanding in students?

Skemp explored this by providing a great analogy of visiting a new town.  He talked about learning the route between two points.  He could get from A to B by following step-by-step directions that were given to him.  Or he could explore the town with the goal of developing a mental map of the town.   In the first case, if he makes a mistake, he will be lost.  He needs to rely on outside guidance to help him get back on track.  In the second case, he can find his way around without having to follow step-by-step directions and can choose from multiple paths.  If he turns the wrong way, he can correct his mistake without any outside help and perhaps learn from the mistake.

How does this relate to math class?

If our students only develop instrumental understanding, they need to remember a series of different step-by-step procedures and when to use them.  They are dependent on the teacher for what to do if they make a mistake.  On the other hand, if they develop relational understanding they can use the connections they have developed to begin to correct their own mistakes and use multiple methods to solve a problem.  Skemp further argues that relational understanding is longer lasting and more adaptable to new tasks.

Thinking about what type of understanding about mathematics we want to foster in our students can help us thoughtfully make choices in our classrooms.  Does everyone understand?

Want to know more?

Read Skemp’s article:

Skemp, R. R. (1976). Relational Understanding and Instrumental Understanding. Mathematics teaching, 77, 20-26.

The Waiting Game: Increasing Wait Time


I’m from New York so it should come as no surprise that I am not very good at waiting.   As a result, I struggled (and still do) to give my students ample wait time during class discussions.

Apparently, I’m not alone.  In 1972, Mary Budd Rowe found that the silence that followed elementary science teachers’ questions and students’ answers lasted less than 1.5 seconds in classrooms.

Why is wait time, or think time as it’s also called, important?

Rowe found that there were many benefits for students when wait time was increased to 3 seconds or more.  These included:

  • A decrease in “I don’t know” responses
  • An increase in the length and correctness of responses
  • An increase in the number of students that participated
  • An increase in student achievement scores

From my experience in math classrooms, I’ll also add that giving students more wait time creates a norm about what is valued in math class.  Solving the problem quickly becomes less important than thinking and reasoning about a problem.  That’s the message I want to give students about math.

So why is it so hard to do?

For me, the silence was uncomfortable at first.  Three seconds feels like forever in a classroom.  I was worried that my students would get off task or become bored.  However, I found that if the questions were appropriately challenging, students would stay on task.  For those who solved the problem in a shorter amount of time, I encouraged them to think of another way the problem could be solved.  After a while, the students and I became more comfortable with the silence and wait time became a norm in our classroom discussions.

I’m curious to learn more about your experiences with wait time.  How long do you wait after asking a question?  What benefits have you noticed as you increase wait time?

Want to know more?  Check out these resources:

Rowe, M. B. (1986). Wait time: slowing down may be a way of speeding up!. Journal of teacher education, 37(1), 43-50.

Rowe, M. B. (1972). Wait-Time and Rewards as Instructional Variables: Their Influence on Language, Logic, and Fate Control.


My Summer Reading List


Ever since I was a child, the summer was my favorite time of year.  One reason was because it allowed me uninterrupted blocks of time to read.  Even today, I keep a list of books I plan to read over the summer.  Since Memorial Day is around the corner, I thought I would share some of the books I have on my summer reading list that relate to research and education.

What’s Math Got To Do With It?  How Parents and Teachers Can Help Children Learn To Love Their Least Favorite Subject by Jo Boaler

I had the opportunity to hear Jo Boaler speak about her research at a conference this year.  A professor of math education at Stanford University, she researches mathematics teaching and learning.  I’m interested to read about how her research informs the solutions she proposes for improving math education.

How Children Succeed:  Grit, Curiosity and the Hidden Power of Character by Paul Tough 

In my last blog post, I talked about the research on grit.  This book explores the importance of character traits like grit and how to promote them in children.  While it’s not specific to mathematics, I’m interested to learn more about why some students succeed and think about how I might apply this to teaching and learning mathematics.

Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States by Liping Ma

This is the only book on the list that I already read but I think it’s worth a reread.  Liping Ma researched the differences between teachers in China and the US and she reports on what she observed.   When I first read the book, I was amazed by how the Chinese teachers understood elementary mathematics in a profound and deep way.  I want to go back and take a deeper look at how these teachers thought about teaching specific concepts.

I’m excited to read these books (hopefully while sitting on the beach somewhere) and I’ll be sure to report back on what I learn after I read them.

What’s on your summer reading list?

What Predicts Success? A Look at Grit.

When Angela Duckworth was a NYC math teacher she noticed something that I think will sound familiar to most teachers: it wasn’t always her most talented students that performed best in her class.

She had a hunch that there was some quality that made those students successful and that doing well in school and life was about more than learning quickly and easily. She left the classroom and set out to research this hunch. After studying what made kids and adults successful in a variety of settings, she found a quality that was a significant predictor of success. She called this grit— a passion and perseverance for long-term goals

Grit was a predictor of which cadets would be successful at West Point, which salespeople would earn the most money, which students would graduate high school, which kids would succeed at the National Spelling Bee and which new teachers in tough neighborhoods would succeed.

I think grit is particularly important in succeeding in math class. I recently worked with a group of fourth graders. When I first began working with them, many students quickly gave up when presented with novel problems that didn’t allow them to follow step-by-step procedures they previously learned. It wasn’t because they didn’t have the ability to solve the problems. It was because they expected to be able to solve math tasks quickly and easily and when they couldn’t, they gave up.

Over time, I tried to create a culture where doing mathematics was not about who completed the most problems in the shortest amount of time. I encouraged them to use their resources to solve the problems and talked to them about how mathematicians often spend a long time solving one problem. After several months, students slowly became more comfortable when presented with challenging problems and the classroom culture began to shift.

The jury is still out on how to best foster grit. What we do know is that Duckworth has found that talent doesn’t make you gritty. In fact, grit is unrelated or inversely related to measures of talent. So how do we best foster grit in our students? I’d love to hear your ideas.

Want to know more?
Find your Grit score and other resources

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?


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

Key words aren’t the key to understanding math

Students in NYC recently finished taking the state math test. As a result, I’ve spent the past couple of weeks watching a lot of test prep going on in classrooms. One of the strategies I see being used over and over is the use of key words.

I admit that I was guilty of using this strategy when I started teaching. I was puzzled when my students could flawlessly perform computations but struggled with word problems. Teaching them to look for key words seemed like an easy fix to this. I had charts in my room listing all the key words and their corresponding operations. Yet, the key words didn’t seem to help them.

So why don’t key words work?

It’s because they don’t allow students to use what they already know to make sense of a situation.

The research backs this up. Drake and Barlow (2007) gave a student the problem below.

There are 3 boxes of chicken nuggets on the table. Each box contains 6 chicken nuggets. How many chicken nuggets are there in all?

Guess what a student who looked for key words answered? 9 chicken nuggets. The student saw the words: “in all” as a signal to add 6 and 3. I would bet that the student could have made sense of this situation and arrived at the correct answer if he drew a picture or reasoned about it. However, using key words led him to an incorrect answer.

Key words encourage students to take a short cut instead of making sense of a situation. If students think about what makes sense, they don’t need shortcuts or key words. They don’t need to worry about what happens when they aren’t any key words or when there are multiple key words in a story.

If we believe that doing mathematics should have meaning for students and make sense to them, then teaching key words doesn’t support those goals. Teaching students to reason about a situation and know why they are performing an operation does.

Have you used key words with your students? What was your experience?

Drake, J. M., & Barlow, A. T. (2007). Assessing Students’ Levels of Understanding Multiplication through Problem Writing. Teaching Children Mathematics, 14(5), 272-277.

Math mistakes: What are they thinking?

I’m embarrassed to admit this, but until recently I was relatively unaware of all the great blogs and websites dedicated to the teaching and learning of mathematics.   I love having so many perspectives on education a click away.

Michael Pershan has a great site dedicated to compiling, analyzing and discussing mathematical errors students make.  In a recent post, he invited a discussion about whether we need to move beyond the identification of misconceptions.

I think we do.

Being aware of the errors is a good first step.  While researchers have done a lot of work on documenting common misconceptions at various grade levels, I don’t think that means that all teachers are aware of their findings.  The learning trajectories I mentioned in my last post are one place you can go to view some of the work that’s been done.

But it’s not enough just to know what the mistakes are.  In order to figure out how to avoid the misconceptions or correct them, we need to know more about what the student is thinking.

The best resource we have in trying to understand our students’ mistakes are our students themselves.  While an incorrect answer may make no sense to us, it makes sense to a child.   For example, I gave the following task to fourth graders recently.

 Which fraction of the region is shaded?


Many students answered one-fifth.   While this solution might not make any sense to us, it made sense to them.  When I asked a student why it was one-fifth, he explained that one box was shaded and there were five boxes in the rectangle.   Although those of us who understand fractions know that fifths have to be equal size pieces, this students was not thinking about the size of the pieces.  He thought about a fraction as the number of pieces shaded out of the total number of pieces, regardless of the size of those pieces.

This is a simple example but it shows that an incorrect answer often makes sense to a student.  It is not simply because he or she made a careless error.  The mistakes students make can often tell us a lot about what students understand and don’t understand.  However, it’s difficult to do this by just looking at the student’s work.  You need to ask the student about what he did.

One of the most useful things I have learned in my Ph.D. program is how to do a clinical interview.   Clinical interviews involve asking a student about what they are doing as they are trying to solve a problem.   You aren’t trying to teach them.  You are trying to get into a student’s head– to see things from his or her perspective.  The goal is to try to understand what the student is thinking, not what you expect or want the student to be thinking.

As they are working on a problem, you can ask them questions like:

  • Why are you doing that?
  • What are you thinking about?
  • How do you know that?
  • Tell me more about what you just did.

There is a temptation to try to lead them to do things differently so they arrive at the correct answer.  However, you need to ask probing questions, listen carefully and try to understand why they are solving the problem the way they are.  The teaching can come after you have an understanding of their thinking.

Now this isn’t something that you can realistically do all the time in your classroom, but I think it’s worth taking some time to interview a student one-on-one in order to try to understand how they are thinking.

Have you tried clinical interviews in your class?  What have you learned?