Tuesday, March 27, 2012

Which tool makes sense?

Confession of a control-freak: I want lessons to run smoothly (exactly the way I envision them). I have written about this issue before and my efforts to give learners more control in the classroom. If the goal of my teaching is learners who possess phronesis, then I need to provide them with ample opportunities to practice making and evaluating choices. In this post, I give another example of my efforts to turn over more responsibility to my learners.

During a recent lesson, my preservice teachers were relearning what it means to add fractions and the role manipulatives can play in supporting understanding. In the past, I would have: 1) put out one manipulative (perhaps the pattern blocks); 2) explained the rules of using the manipulative (2 yellow hexagons represent a whole); and 3) provided them plenty of practice in using the manipulative to represent fraction addition. Then, I would have them put away the pattern blocks and grab some fraction circles and go through the process again. As I said, controlling. I came to realize that my management of the tools and the rules was disempowering my learners. This time I simply put out a variety of manipulatives and asked them to explore.

Schema Activation: All learners need time to explore the tools at their disposal. Please take five minutes to play with any of the manipulatives at your table.

I find that it is important to provide learners, regardless of their age an opportunity to play with manipulatives (they are going to anyways, I might as well embrace it). The learners spend the time building, organizing, and comparing the shapes. At the end of five minutes, I ask them to put the manipulatives back into their separate containers. This provides a break that I find helps learners to shift their vision of the manipulatives from toys to tools. I make sure to make this point explicitly.

Focus: Each of you is now going to pick a manipulative and consider what happens when you add two unlike shapes. For example, If I combine the blue rhombus and the red trapezoid from the pattern blocks, then what would I get? How did I get it? Why does it work? When does it work? And what if I combine other shapes, will my thinking still hold or do I need to adjust it?

This is usually where the manipulatives' rules are shared. Instead, I want to provide a framework of questions for them to keep in mind as they consider combining shapes. Not having done this workshop before, I am unsure this will provide them with enough structure. If there is any uncertainty, I am prepared to model how my wife had thought of combining the two pattern blocks by focusing on their side lengths: 4 units (rhombus) plus 5 units (trapezoid). There is no need, however, as the learners get right to work.

Activity: Learners work in groups on the task.

This is an opportunity for me to conduct some formative assessment. As I walk around, I try to focus on asking questions that check for understanding - any teaching/leading questions can wait until later. Based on the data I gather, I organize the remaining part of the lesson.

One group is working together with a single manipulative. They are discussing different ways to view the result. Another group has split the manipulatives between them and are seemingly working separately. But every so often, one learner shares with the others her conjectures and the others provide their feedback. From these observations, I ask two learners if they are willing to share their thinking with the rest of the class. They agree.

Reflection: Mathematician's Chair - learners share their thinking and any struggles with the rest of the class in order to expose their work to the larger learning community.

The first person I ask to share was working with her group to combine two shapes from the fraction circles. They had chosen to combine the two pieces shown at at the right. These seem pretty basic but the group had gotten into an interesting discussion about what represented the whole. The gist of the discussion, which she recounts for the class, was, "If we let the white circle be the whole, then the result is one-and-a-half white circles. If the orange half-circle is the whole, then the result is three orange half-circles. It all depends on what you denote as the whole."

Next, a learner shares what she found using the pattern blocks. It goes something like this. "I found it easiest to break down each shape to the smallest shape. So the blue rhombus is two green triangles and the red trapezoid is three green triangles. That means together they are five green triangles."

At this point, another member of this learner's group interjects. He had taken the idea and tried to apply it to the Cuisenaire Rods. He essentially points out, "It really comes down to names. I can break down any of the sticks to the smallest one, the white block, and then combine them because they have the same name."

Teacher's Reflection on the Workshop
I am pleased with the workshop. The learners were able to recognize the importance of identifying a whole and the need to find a common name (denominator) in order to combine fractional pieces. These are big ideas that are often reduced to rules without reason for students being instructed about fraction addition.

There are a few things I will do differently next time. First, I need to make even more explicit my decision to give them choice about which manipulative they would work with. Much of the work of teaching is invisible and I worry this was the case in this workshop. Next, I want to have a larger discussion about the wise use of manipulatives in math class. This might be a conversation around Deborah Ball's article, Magical Hope. Finally, I want to further expand on the idea raised during the reflection regarding naming. Using the Cuisenaire Rods, we could explore different names such as those shown below.

While this lesson did not go perfectly (they rarely do), I see real progress in my learners and myself. This experience will provide a pivotal experience that we can return to in subsequent lessons as we talk about appropriate use of educational resources. It also provides data that I can use in future classes. It is impossible to say how learners will respond, but now I have examples of how learners have responded. If necessary, I can always share these examples with as models of how others thought about the task. Often, one of my most effective teaching moves is to ask, "Would you like to see how other learners thought of this problem?"


  1. At the beginning of my teaching career, I used manipulatives as you did in the past. I explained the rules and then students practiced my rules. Of course, this wasn’t effective and, for the most part, I stopped using manipulatives in my classes.

    With the implementation of a new curriculum, I gave them another try. Had to. The phrase ‘concretely, pictorially, and symbolically’ was now part of many learning outcomes (as opposed to just being a suggested strategy as in the previous curriculum).

    This time, I gave students time to explore. Instead of giving students questions like ‘Model 1/2 plus 1/3 using pattern blocks’ and guiding them through converting a trapezoid and rhombus (the hexagon was usually equal to one in my class) to triangles, I changed the question as you have done. (What might a trapezoid and rhombus add to? Why?) What a huge difference. It was easier to help students make the connection to lowest common denominators. In the past, I think my students saw using manipulatives and LCDs as two separate strategies, no matter how much I tried to connect the two. A bit of advice I was given helped me make this change: the overhead pattern blocks should only be used by students as they volunteer to share their thinking.

    To introduce pattern blocks, I also take advantage of the ‘they’re going to play around anyways’ idea. I ask them to build a shape (without using the square or tan rhombus). Next, I ask them to build the same shape using different blocks. I ask them to calculate the value of their shapes if the blue rhombus has a value of 2 (or 1/3, or 1, or 1/6 for that matter). Students/teachers are now ready to list pattern block properties. We discuss why I asked them to exclude the square and tan rhombus but also how they are related to the others. (Not an original idea here – I just can’t remember the source.)

    Thanks for sharing this. I enjoy learning from your blog.

    1. Thanks for your comments - especially since they seem to support what I've written. One of the goals for this blog is to share ideas and get feedback. It is through constructive criticism I hope to hone my understandings. Consequently, your response detailing similar experiences and adding new ideas really helps me.

  2. I think it's natural for a teacher to be a "control freak" when it comes to our lessons since we'd work hard planning for it and we'll be damned if it doesn't go "exactly the way I envision them." But I'm getting better at letting go, giving minimal instructions, and when I do I'm rewarded with a lot of good math conversations among the students.

    I really like what you'd done here, and I love your Focus section above because we do have to rein in the exploration phase, it's good to be explicit here. I have the kids play with pattern blocks too for fractions and ask something like this: 1) If two green triangles equal 1 square unit, then what is the area of one red trapezoid? 2) If a yellow hexagon is 3 sq units, then what is the blue rhombus?


    1. Getting out of the way of students' learning is difficult for many teachers, but I've found it extremely rewarding. I think it was Debbie Miller's book, "Teaching with Intention," that first got me to really consider the importance of trusting my students. Of course it helps me to trust when structures (like the focus - great observation) are in place to increase the likelihood of success.

      Thanks for your comment, Fawn.