Tag Archives: student thinking

Using manipulatives to build on what students can do

In my last post, I talked about building on what students can do instead of focusing on what they can’t do.  One way many of us try to do that is by using a variety of manipulatives in our classes.

I love using concrete objects with students of all ages.  But I can remember several times when other teachers or administrators made fun of me for using “toys” with my 8th graders to teach algebra.

Manipulatives can be a great way to provide access into lessons for all students and when used well they can help build deep understanding.

But as two researchers, Stein and Bovalino (2001) pointed out: “manipulatives do not magically carry mathematical understanding.”

They identified two problems that can occur when using manipulatives.

1.  They can sometimes be used in a way that only requires students to mindlessly follow what they see the teacher do.   As a result, students mindlessly move around blocks just as they would mindlessly use a formula without meaning.

2.  They can lead to unsystematic and non-productive exploration.  If the mathematical goal is not clear and the activities are not carefully designed, the manipulative does not serve as a tool for developing the concept the teacher intends.

The researchers contend (and I would agree) that manipulatives need to be used in a way that links an activity a student can do to a new concept by allowing them to take in new information and give it meaning.

I’ll give an example.  I recently worked with young students on how to find the area of rectangles.  I gave out square tiles and asked them to find the area of a rectangular region.  All of the students could complete the task by simply covering the region with the square tiles.  What did they learn by doing this?  Not much.  At this point–they haven’t learned anything new. 

It wasn’t until I asked them to how to find the number of squares without counting all the tiles that they began to develop new understandings.  Some counted the number of rows and the number of tiles in one row and multiplied.  Others counted the number of columns and the number of tiles in one column and multiplied.  Others used repeated addition. As we worked on different sized regions and shared different strategies, they began to develop a rule for finding the area.  Eventually, they were able to find the area when the dimensions were given without using any tiles.

Manipulatives allowed them to access the activity and have a concrete way to justify their answer but using the manipulatives alone did not lead to the new understanding developed.

If the student could do the activity before the lesson, he or she did not learn anything.   Learning occurs when students use what they can do to develop a new understanding.

Or as the researcher Doug Clements (1999) says:  “Students may require concrete materials to build meaning initially, but they must reflect on their actions with manipulatives to do so.”  Encouraging that reflection is a difficult task and one that I think often goes missing from lessons.

How do you use manipulatives?  How do you encourage your students to reflect on their work with them?  I’d love to know.

What to know more? Read what the research has to say:

Clements, D. H. (1999).  Concrete manipulatives, concrete ideas. Contemporary Issues in Early Childhood, 1, 45–60.

Stein, M. K., & Bovalino, J. W. (2001). Manipulatives: One piece of the puzzle. Mathematics Teaching in the Middle School, 6, 356–359.

The worst thing you can say to a student struggling in math

“It’s easy.  Let me help you.”

Ok.  Maybe that’s not the worst thing you can say to a student, but it’s up there. Yet, well-intentioned teachers and parents say it all the time. Why?

For starters, the problem is easy to us. We don’t understand why a student is struggling with something that seems so obvious to us. Second, we are often uncomfortable watching students struggle and sometimes, the quickest way to alleviate the discomfort is to show a student how to solve the problem.

So why should we avoid saying this to students?

It’s likely that a student could interpret it as: “If it’s easy and I can’t do it, I must not be very smart.”   This is the last thing a struggling student needs to hear.

In addition, offering help creates the misconception that math is not something students can make sense of on their own. Furthermore, the help you give may not address the real reason the student is struggling.

So what can you do instead of offering to show a student how to solve a problem? The research offers one solution– find out why the student is struggling by interviewing them.

“Tell me what you are thinking,” can be a good place to start. Maybe the student doesn’t understand the vocabulary or language in a problem. Or maybe the student is missing the prior knowledge that is needed in order to solve this new task.

Finding out why the student is struggling is a good first step. We’ll explore next steps in future posts.

Want to know more?

Take a look at chapter 3 in Van de Walle’s book: Elementary and middle school mathematics: Teaching developmentally.

Read Children’s Mathematics: Cognitively Guided Instruction

Check out this article about how one teachers tried this in his classroom:  Buschman, L. (2001). Using Student Interviews To Guide Classroom Instruction: An Action Research Project. Teaching Children Mathematics, 8(4), 222-27.

What matters when choosing tasks for students?

“Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.”  -Stein, Smith, Henningsen, & Silver, 2000

What are we doing today?

One of the biggest decisions we make as teachers is choosing the tasks our students will work on during class.  We use tasks from textbooks and the Internet.  We borrow tasks from colleagues or design our own.   No matter where the tasks originate from, the nature of tasks we select affects our students’ ideas about what it means to do mathematics.  Different tasks provide different opportunities for students to learn mathematics.

One way to think about how to select a task is by looking at the cognitive demand required by it.   Cognitive demand refers to the type of student thinking required to solve the task.   A group of researchers developed different categories of cognitive demands found in mathematical tasks.

Low-level cognitive demand tasks can be solved by memorization or by using procedures that don’t have meaning for students.  High-level tasks engage students in using procedures with attention to the reasoning behind them or in what they label “doing mathematics”–making conjectures, justifying, and interpreting.

After studying over 500 tasks in middle schools, the researchers found that the greatest gains in student achievement occurred in classrooms where teachers used high-level tasks and the cognitive demand of the task was maintained as students worked on the tasks.   It was also noted that although many tasks started out requiring high cognitive demands, the demands of the tasks often decreased as they were implemented in classrooms.

Of course, there is a lot more to think about when selecting tasks.  How do the tasks build on what your students already know?  How do the tasks engage students?   What particular mathematical concepts are fostered by the task?  But considering the level of thinking you want to foster in your students is a good place to start.

Want to know more?   Check out the research below.

Stein, M. K., & Lane, S.  (1996).  Instructional tasks and the development of student capacity to think and reason:  An analysis of the relationship between teaching and learning in a reform mathematics project.  Educational Research and Evaluation, 2(1), 50 – 80.

Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. Teachers College Press, 1234 Amsterdam Avenue, New York, NY 10027.