Tuesday, April 24, 2012

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.

Tuesday, April 17, 2012

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?

Thursday, April 12, 2012

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?

Thursday, April 5, 2012

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?"