Tag Archives: learning

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.

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


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.