Tag Archives: student learning

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.

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.

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.

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.

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?