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.

Workshop

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?"

What did you see/hear?

Last week, I attended the Michigan Reading Association's Annual Conference where I went to a session lead by Doug Fisher. He was presenting on Response to Intervention from a gradual release of responsibility perspective. It was the gradual release model that I was most interested in - especially since his book with Nancy Frey presents a slightly different version of this approach than the one I use. But I will leave that discussion for a later post. Today, I want to share my current thinking about what makes for an effective demonstration lesson (what Doug labels Purpose & Modeling).

From Doug Fisher's Michigan Reading Association Presentation

The demonstration I am sharing comes from a workshop for my Teaching and Learning Middle Grades Mathematics [TLMGM] course (a combination content and methods course that our secondary majors take prior to student teaching). This workshop focuses on comprehending the purpose behind a lesson that a teacher might encounter in an unfamiliar curriculum. Many of our student teachers find themselves using curricula that typically require a significant amount of professional development. Because our student teachers have not had this support, they sometime struggle to implement the lessons effectively. Therefore, I hope to share an approach I might use to better understand the rationale behind a lesson.

I used the ShowMe App to take a picture of a number string from a lesson found in the Context for Learning Mathematics series and then added my thinking. ShowMe records both my whiteboard annotations and voice to create a video that I can share with others.

A key piece to any demonstration is the debriefing: What did you see and what did you hear? This reflects Cambourne's perspective on who is ultimately responsible for learning: "Learners need to make their own decisions about when, how, and what 'bits' of information to learn in any learning task." By asking the learners what bits of information they attended to during the demonstration, the teacher is gathering important data that can inform future instruction (formative assessment). If the observers did not attend to something important presented in the demonstration, the teacher can highlight the missing points by saying, "I noticed that I was also..." The resulting list provides an anchor chart that learners can refer back to as they take more responsibility for employing the approach.

After I shared this demonstration with my learners in TLMGM, they noticed that I was trying to understand the string by:

• making connections between the expressions;
• considering different representations that might further my understanding;
• recognizing that computing the answers might help but wasn't enough;
• thinking about ways to put the expressions into a context; and
• analyzing my options before jumping into any plan.

Satisfied that they had attended to the major points of the demonstration, I provided them with further resources related to the Ratio Table unit and had them work collaboratively. They tried to look at the strings from the various angles in order to understand the purpose of the lessons.

During the workshop's reflection, I provided the opportunity for them to share what they had uncovered through Twitter. This provided everyone who wanted to contribute a voice and me with another artifact that I could use for formative assessment. Here is a sample of the discussion:

UPDATE 3/21/12
Reflecting back on this post, I have come to realize that if I were going to embrace the idea of "flipping" my class, this is what it would look like. I would use ShowMe to create demonstration lessons and have my learners watch them. When they finished, I would ask them to Tweet those things that they noticed in order to assess what they paid attention to and create an anchor chart for the class.

I know that my learners (college students) may differ from your's if you teach in a K12 school. But what do you think? Would this work for you?

How's it going?

When learners enter my class the first day of the semester, they typically see the following projected on the front board:

Sooner or later, the cups draw a learner's attention and the question is asked: "What are these for?" Which is closely followed by, "Is it some sort of stop light?"

Indeed, three cups are stacked in the midst of each table group - a green, a yellow, and a red cup. And they do represent a sort of stop light but it is for me, not for them. I explain that when a group is showing green I interpret that to mean that the group is making good progress. If yellow is showing, I know that they have a question but that it isn't stopping them from moving forward. I try to get to these groups when I can, and sometimes they work out the issue themselves. Red means that a group is stuck and needs help in order to get moving again. Connecting with them becomes a priority for me. Some teachers use index cards instead of cups, but I like the cups because I can hear them being re-stacked when a question arises - even when my back is turned.

Early in the semester, learners will often raise their hands without changing the cups. I always try to ask, "What kind of question is it: red or yellow?" This helps them to be metacognitive and not let every little question sidetrack their efforts. A student teacher told me how using the cups in his class significantly reduced the number of needless questions being asked. The kids told the student teacher that when they took the time to consider their questions often they realized they could answer the questions themselves.

I even have my learners use "red cup" or "yellow cup" in the subject lines of their email questions to me so I know what kind of priority I ought to assign each emails. But I was a bit mystified when I got an email from a recent graduate who had written "Green Cup" as the subject.  It read:

Hi Dave! 

I was in the middle of grading quizes during my planning period and thinking about my todo list for the end of this first grading period and I was struck by this thought. I feel confident, calm, and content with how this semester is going. Granted, there are many things I already decided to change for next year. However, I do not feel a great deal of stress or anxeity. The COE and student teaching was an intense program and very stressful but it wasn't until just now that I realized how much I learned and how well it prepared me to have a classroom of my own. There are still things (like classroom management) that I am continuing to work on, but for the most part everything is going very well. I just wanted to say thank you for the support you provided though out my time in the COE. 

Hope all is going well.

Green cup! 

How do we make thinking visible?

Consider how often what we learn reflects what others are doing around us. We watch, we imitate, we adapt what we see to our own styles and interests, we build from there. Now imagine learning to dance when the dancers around you are all invisible. Imagine learning a sport when the players who already know the game can't be seen. Bizarre as this may sound, something close to it happens all the time in one very important area of learning: learning to think.

Thinking is pretty much invisible. To be sure, sometimes people explain the thoughts behind a particular conclusion, but often they do not. Mostly, thinking happens under the hood, within the marvelous engine of our mind-brain. 

Not only is others' thinking mostly invisible, so are many circumstances that invite thinking. We would like youngsters, and indeed adults, to become alert and thoughtful when they hear an unlikely rumor, face a tricky problem of planning their time, have a dispute with a friend, or encounter a politician's sweeping statement on television. However, research by our group and others shows that people are often simply oblivious to situations that invite thinking. …

These paragraphs come from the paper, Making Thinking Visible (PDF), by David Perkins. It provides the backdrop for a session on Metacognitive Memoirs that I am co-facilitating with John Golden for the Michigan Reading Association 2012 Conference. (handout)

Schoenfeld's research (PDF) demonstrates some of the invisibility associated with solving problems in mathematics. He finds that students often get stuck when trying to solve true problems (as opposed to exercises they can complete with some predetermined procedure).

Compare this with an expert who constantly shifts perspectives in order to ensure continual movement toward a solution.

What Schoenfeld determines is that by modeling for students how an expert problem solver thinks, the students can learn to monitor their thinking while solving problems and make the necessary adjustments to be successful.

In the MRA session, we look to children's and young adult literature for examples of characters involved in solving problems. These become models that students can refer back to when it comes time to problem solve. Sometimes the thinking in a book is visible, as in this case from Fish is Fish. (I was introduced to this excellent example by the book, How Students Learn.)

Not all authors make the thinking of their characters visible, however. Therefore, we ask participants to use the comprehension strategy of making inferences to highlight the thinking associated with problem solving. Keene and Zimmermann suggest that effective readers (from Mosaic of Thought, Second Edition - page 260):

• use their schema and textual information to draw conclusions and form unique interpretations from text
• make predictions about text, confirm their predictions, and test their developing meaning as they read on
• know how to use text in combination with their own background knowledge to seek answers to questions; and
• create interpretations to enrich and deepen their experience in a text.

So with a stack of sticky notes and a selection of books, the participants consider how to infer the thinking of various characters - especially during critical problem solving situations. Using reader's chair, participants then share their inferences and what these inferences suggest about the thinking associated with problem solving in the stories.

Hopefully, this flows into the writing of metacognitive math memoirs (which are discussed here and here).

When were we engaged?

Teach an engaging lesson and kids are engaged for a day. Teach them how to engage and they are set for a lifetime.

Teaching students how to engage entails: (1) identifying what engagement looks like; (2) setting the conditions under which engagement can occur; and (3) providing opportunities for them to reflect on their engagement. I described the first point here and the second here. In this post, I address the third point.

Confessions of a "bad" math teacher: I often confuse engagement with conformity. In the past, I have given a participation score based on how engaged my students were in doing the assigned work. If they did it all, then they got 100%. If they did half the work, they got 50% of the points. I have come to understand, however, that participation is not necessarily engagement. The Engagement Continuum shown below highlights the problem. While the students may have demonstrated interest in getting all the participation points, there is no evidence that they were truly engaged in the content.

Instead of offering participation points (faux engagement), I now ask students to demonstrate and reflect on their ability to engage. At the end of the semester, they turn in a predetermined number of workshops that provide evidence that they know what it takes to engage themselves. Their engagement score is dependent on their ability to identify their level of engagement and explain how they achieved that level.

Here is an example from this semester. At the end of our first unit, we make anchor charts about what it means to do mathematics. (I have written about this before.) I rarely give them enough time to complete their chart - just enough to have something to share with their peers. This is a draft completed by a group trying to use a blueprint for a school as an analogy for what we have been discussing in the unit.

After the class, the group stayed to complete their anchor chart. They didn't have to do any more work on it, but they wanted to finish it. It was interesting to listen in as they continued to attend to small details and talk to one another about their vision for their teaching.

One of them asked me if he could use this workshop as an engagement exemplar. I told him that any workshop, in class or assigned as homework, was available.

It was clear to me that this group was committed to completing the task, but where it fits on the Engagement Continuum is up to them to demonstrate. Only they know what was going on in their minds as they worked on this project. I am through trying to attach my own judgements regarding engagement to students' efforts. They need to be able to identify when they are engaged and determine how to replicate the conditions that led to their engagement.

I must say that I am interested in what they submit for their engagement exemplars - much more interested than when they simply recorded how many of the workshops they had completed. It is sort of funny; this approach has increased my engagement as well.