Tag Archives: teaching

How do you choose what model to use when teaching fractions?


The three representations above can all be used to model two-fifths.   Does it matter which one you choose when introducing fractions?

In my last post, I talked about why concrete objects or models can be helpful in building on what students can do.  What I didn’t talk about was how to choose a model when trying to foster a particular mathematical concept.

Fractions are a a good example since most teachers and researchers (Cramer & Henry, 2002; Siebert & Gaskin, 2006) agree that models are important in helping children develop fraction concepts.

But how to you know which one to choose?  Do you use pizzas, fraction strips, cuisenaire rods, number lines, chips?

The first thing to note is that different models offer different opportunities to learn. Let’s look at the three main models used for teaching fractions and what they offer:

  1. An area model, like fraction bars, can help students visualize parts of a whole.
  2. A model with discrete objects, such as counters or chips, can help students begin to think about fractions of a set.
  3. A linear model, like a number line, can help students see that there is always another fraction between any two fractions.

As Susuan Lamon points out in her book Teaching Fractions and Ratios for Understanding: 

“No one model is a panacea; every model has some useful features, it wears out at some point and it is up to the teacher to use it wisely.  If you have to spend valuable class time teaching students how to work with the model, it is wasted time that is not being used to teach fractions (p. 149)”

Contrary to what many outside the classroom believe, this confirms one of my basic beliefs about teaching:  It’s complicated.  There is no one right answer for how to do things.

Yes– it matters which model you choose.  Each model will foster certain understandings about fractions and mask others.  That’s why it’s important to be aware of the advantages and disadvantages of the model you choose.   That’s also why it’s important to expose students to more than one model.

But it’s not a simple matter of saying this is the right model or this is the wrong model or make sure students can use all of them.

Choosing which model or manipulative to use depends on your instructional goal.  If the goal is to work on partitioning a unit, an area model might work well.  If the goal is to work on viewing a fraction as a quantity, a linear model might work well.  If you have identified that students are struggling with fraction of a set, you may want to introduce a discrete model.  Being clear on your goal and the advantages and limitations of each model will allow you to choose the right model for you and your students.

There may not be one right answer when it comes to choosing a model for teaching fractions, but the choice does matter.

What about you?  How do you choose models for teaching fractions?

Want to know more?  Check out the resources below.

Cramer, K., & Henry, A. (2002). Using Manipulative Models to Build Number Sense for Addition of Fractions. In B. Litwiller & G. Bright (Eds.), National Council of Teachers of Mathematics 2002 Yearbook: Making Sense of Fractions, Ratios, and Proportions (pp. 41-48). Reston, VA: NCTM.

Lamon, S. J. (2005). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. Psychology Press

Siebert, D., & Gaskin, N. (2006). Creating, Naming, and Justifying Fractions. Teaching Children Mathematics, 12(8), 394-400.

The “Math Wars” still rage on

What amazes me about working in schools is that when I walk into one I am immediately transported back to being a student.  The clothes may be different and the hairstyles may have changed, but in the over twenty-five years that has passed since I was a student in an elementary school the model for teaching math still looks very similar in a lot of classrooms.

The teacher stands at the board (maybe now it’s a smart board), models an algorithm and the students practice it at their desks.  Now I’m not saying that this happens in every school.  I’ve had the chance to visit classrooms that use very different approaches and I like to think that when I was a teacher, I pushed for an inquiry-based way of teaching as much as I was able to at the time.

But it’s perplexing to me that 25 years of research haven’t been able to change the traditional model in a lot of schools.  Imagine if the hospital I went to 25 years ago still used the same methods today.

However, after reading the recent op-ed piece in the New York Times that argues against math education reform, I see why it is so hard to enact change.   There are still many people who don’t buy into new methods of teaching math, hence the so called “math wars.”   The fact that this debate between traditional and reform methods of teaching still wages on is shocking to me because we know that our traditional ways of teaching are failing lots of students.

After reading the piece in the Times, you should check out Professor Keith Devlin’s thoughtful paragraph- by-paragraph critique of the article.   It makes a number of good points that I won’t rehash here.

The thing that bothered me most about the op-ed was that there was no connection to what we know about how students learn.  There is no attention given to what the research says about student learning.   There has been plenty of work (see the references at the end for some of it) that demonstrates that students learning math in reform-based classrooms outperform students in traditional classes and these students report stronger motivation and interest in math.  What’s more, the reform approach has been shown to be successful with students of diverse backgrounds.

It just doesn’t make sense to me to keep doing what we’ve been doing and expecting different results.

What do you think?

Want to know more? Check out some of the research below.

Boaler, J. (2002). Experiencing school mathematics: Traditional and reform approaches to teaching and their impact on student thinking. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

Knapp, M. S., Adelman, N. E., Marder, C., McCollum, H., Needels, M. C., Padillia, C., Shields, P. M., Turnbull, B. J., & Zucker, A. A. (1995). Teaching for meaning in high poverty schools. New York: Teachers’ College Press.

Silver, E. A., & Stein, M. K. (1996). The QUASAR project: The revolution of the possible in mathematical instructional reform in urban middle schools. Urban Education, 30, 476–521.

Van Haneghan, J. P., Pruet, S. A., & Bamberger, H. J. (2004). Mathematics reform in a minority community: Student outcomes. Journal of Education for Students Placed at Risk, 9(2), 189–211.

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.


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.

Learning Trajectories: A research-based tool you can use right now


In an earlier post, I talked about the gap between the work being done by researchers and the work being done by teachers in classrooms.  Today, I’d like to talk about a resource that I think is a good example of a tool that connects research and practice.

The Common Core Standards have received a lot of attention lately.   I often struggle when trying to unpack the Standards in order to develop a lesson.  A research team from NC State has created an interactive tool, available at http://www.turnonccmath.net/, which I have found very useful in designing lessons.  The researchers developed 18 learning trajectories for the K-8 Common Core Standards for Mathematics.   These learning trajectories describe how students develop an understanding of a particular concept, or set of concepts, over time.

The hexagon map allows you to click on one of the trajectories or on an individual standard and get detailed information about how students might move from prior knowledge and informal ideas to sophisticated understandings of various concepts.  Also included throughout are common student misconceptions and a variety of models and representations that have been shown to foster particular understandings.   What I love about the trajectories is that they include the research that was used to build them.

So how do I use them?  Recently, I needed to prepare a lesson for fourth graders on angles so I went to the site and clicked on the Shapes and Angles Trajectory.  I was able to view suggested activities, such as the angle game where students stand and follow directions to rotate a half turn or quarter turn.  I also read about common student misconceptions, such as the fact that students sometimes think that an angle with longer rays is larger than an angle with shorter rays, and viewed questions that assess this misconception.   The references were also included in case I wanted to go back to some of the original papers and explore any of the ideas further.

Of course, it wasn’t the only resource I used when I was preparing the lesson.  I used my prior experience of teaching angles and other resources I have collected over the years.  But it was nice to have place to go that summarized the major research in a way that allowed me to think about how the students might develop a concept and the challenges they might face along the way.

I’d love to know of any resources you use that connect research and teaching.