The Wumanians are aliens?

After providing my preservice elementary teachers with this coffee-stained Rosetta Stone, and giving them time to try to solve the place value problem, I begin pulling out other artifacts found in the Wumanian student's lab. These are intended to reinforce the efforts of those learners who are on the right track and redirect those who are still floundering. My rule of thumb is to pull out a new artifact whenever a learner asks, "Is this right?" (Please don't give away my little secret.) The idea is that the artifacts will enable them to monitor their own progress without relying on me to validate their work.

I might start with something simple like the ruler shown below. Except, I do not tell them its function - only that it was found in the lab. Therefore, it contributes to the mystery while reinforcing the place value emphasis in the Wumanian system.

The next artifact would be the clock. It does not add anything new to the exploration of the system. However, it does represent a familiar object to which they can connect. It suggests that their prior knowledge and experience will be of help in solving this problem.

The calendar offers the most support in completing the numeric portion of the Wumanian system. While it is a familiar object to the preservice teachers, its structure continues to hide some of the patterns necessary to completely understand the place value system. In other words, it does not completely solve the problem for them.

When the preservice teachers see the cards and the coins, the alien nature of the story becomes clear. One of the hardest things to avoid is constantly trying to translate from our system to the Wumanians' system. I believe this part of the story allows them to look at the new system from a different perspective. These artifact also provide certain clues to the structure behind the system and the numerals.

Finally, I introduce the lyrics to a song using the Wumanian words associated with the numbers. I have considered including a recording of the Wumanian student singing the song. The repetition in the language is another pattern that helps the preservice teachers to complete the Rosetta Stone worksheet. It also highlights one of the problems with the language associated with our system.

Once the preservice teachers have broken the code for the Wumanian system, we begin to consider how Wumanian children might learn how to add, subtract, multiply, and divide in their new system. This provides me the opportunity to introduce centers that might be used to develop operational fluency. A few of these centers will be shared in the next post.

What comes after erwu-si?

In last week's post, I introduced the story of Wumania's struggle to find an efficient number system. This context, used during the first few weeks of a mathematics education course for preservice elementary teachers, provides learners with a fresh perspective with which to explore place value concepts. The previous lesson asks learners to predict the number system developed in Wumania given that it uses only the symbols found on their flag.

Problem Solving Workshop

Goal: The learner will use patterns found in various representations to identify the underlying structure of an unknown number system.

Scheme Activation: Sharing Our Predictions

Learners post the number systems they developed using the five symbols.

As the examples above show, the systems they create typically reflect an additive model - like those found in ancient cultures. 

Focus: Problem Solving (from NCTM Process Standards)

• Build new mathematical knowledge through problem solving;
• Solve problems that arise in mathematics and in other contexts;
• Apply and adapt a variety of appropriate strategies to solve problems; and
• Monitor and reflect on the process of mathematical problem solving.

Activity: Part of the Picture

I introduce an artifact of the Wumanian number system with the following story:

Good news! We have found more artifacts from the lab of the student who solved the number system problem for Wumania. As they are cleaned and catalogued, they will be made available to you. Maybe the most exciting is this sheet containing various representations of the system. Unfortunately, there seems to have been some sort of accident in the lab. A major portion of the sheet is covered by what we are assuming is a coffee spill. Still, this is an amazing find and ought to support us in our quest to understand this new Wumanian number system.

[I created the artifact based on the ideas presented in the article, Using Language and Visualization to Teach Place Value. It is meant to immerse learners in multiple representations. This allows them to choose which information to focus on. Some learners focus on the patterns vertically, in a particular column, while others look for relationships horizontally.]

Reflection: Recount

• What did you do?
• So what new knowledge did you build?
• Now what problem solving strategies might you apply to make further progress?

How old would you be on Wumania?

In this post, I want to share a modified version of a home workshop that I assign in our second mathematics education course for preservice elementary teachers. This is the first in a series of posts around the number system used in a fictional land called Wumania. I was reminded of how much fun I have with these lessons after watching this video of my former colleague, Stephen Blair. If you enjoy the activity and want to share your ideas, please do so in the comments.

Flag of Wumania

Connection Workshop (Wumania)

Goal: The learner will attempt to clarify the meaning and purpose of a number system by making connections.

Schema Activation: In your journal, make a list of the places where we encounter numbers in our lives.

Focus: Making Connections

• Effective readers clarify and give purpose to text by connecting to relevant, prior knowledge.
• Read the article, Developing Number: What Can Other Cultures Tell Us?, keeping track of connections to what you know about early number concepts.

Activity: A History of Wumania's Number System

There is a civilization far, far away called Wumania. In fact, it is so far away that it has had no contact with our civilization, except for this one story that I heard. The story goes like this:

When counting fingers, if a Wumanian held up no fingers, she would say, “na,” what we would call zero. For one finger, she said the first letter of their alphabet – in our case that would be A. Two fingers were B. Three fingers were C, and so on. The problem was that when they got to their Z, they had run out of letters and therefore out of numbers. Anything more than Z was “many" to Wumanians.

The people of Wumania were very dissatisfied with their own number system. In fact, the Wumanians were so dissatisfied they offered all of their students generous grants to work on designing a different number system. Wumania's students worked on this problem for several years until one day a certain student announced the successful design of a new system. The new system, the student said, would use only the five symbols from the Wumanian flag:

Furthermore, the language representing the numbers would be comprised entirely of combinations made from the following words: na, yi, er, san, si, and wu.

The only problem was that before the student was able to publish a full description of the number system, she disappeared. But she left behind the different sized blocks that she was going to use to explain it.

Your task is to come up with a possible number system for Wumania that fits the description. It might help to answer these questions:

1. What does each symbol represent?
2. How would you use your system to count from our "zero" to "twenty" in Wumanian?
3. How old would you be on Wumania?

Reflection: Evaluating the Wumanian Number System

• Why do you think the X-Manians were dissatisfied with the original system [A, B, C,…]?
• Is the system you inferred that the student designed better than the original system? Why or why not?
• What is the least amount of information that you would need about the new system so that you might more accurately represent it?

When is it okay to use a calculator?

All too often, I run into teachers (both preservice and inservice) lamenting that kids are using calculators to compute something simple, like 6 x 7. These teachers express their frustration by threatening to not let kids use any calculators until the kids prove that they know their facts. And there will be no calculators for any simple computations. I understand this thinking but I am not sure it will achieve the desired result - wise calculator use (i.e. phronesis).

Problem is, if teachers are the ones deciding when their students can or cannot use calculators, then students are not able to practice this critical-thinking skill for themselves. Consequently, whenever a calculator is available in the future it can be used because that was what the students learned in school. As an alternative to controlling calculator use, I suggest a couple of activities that can help students to decide for themselves when it is appropriate to reach for the calculator.

The first activity comes from the article, John Henry - The Steel Driving Man. This article suggests several different experiments involving routines that can be completed by human or mechanical effort (eg: sharpening pencils). Students are asked to predict which method will take longer, gather data, and compare the results using box-plots. 

The experiment I am most interested in involves computing multiplication facts with and without a calculator. I give pairs of students four worksheets like the one shown below.

Each of the four worksheets is different. As one student completes the worksheet, the other one uses a stop watch to time the effort (errors add an extra five seconds to total time). This is repeated until each student completes two sheets - using the calculator for one but not the other. 

Classroom data are gathered and typically show that the students complete the worksheets quicker without a calculator. Discussing the results can provide students with an opportunity to reflect on whether or not it is efficient to use a calculator for basic facts. In the few cases when it is faster to use a calculator, the issue is usually that the student has a lot of wrong answers when computing without a calculator. Teachers must decide an appropriate course of action in these instances.

A second activity that I use to develop students' wise use of calculators involves a worksheet of multi-digit multiplication items. Instead of assigning the entire worksheet, I ask students to pick four items to compute without a calculator, four items to estimate, and four items to use a calculator on. The students are also expected to explain why they selected the approach to use with each item.

Calculator phronesis, the wise use of calculators, requires opportunities for students to experience activities that involve metacognitive aspects. We teachers will not always be there to guide students' choices, but this is not to suggest that we do not have a responsibility to help students to develop this ability. It is my hope that through these classroom experiences, students will be able to ask and answer for themselves the question, "When is it okay to use a calculator?"

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!