Can you help me with subtraction?

[Note: Based on feedback from readers, it seems this post needs some context. This is part of a series on exploring a new number system in a course for preservice elementary teachers. The purpose of this unit is to provide these future educators with an experience of what it is like to learn concepts of number and operation using a conceptual approach. In this final post of the series I describe the days leading up to the unit's final assessment. You can see the entire series by clicking on the Wumania tag at the bottom of the post.]

As we near the Wumanian number system assessment, some of the preservice elementary teachers express concern about their ability to demonstrate their fluency in computing with multi-digit numbers. I suggest that they consider putting the expressions into context using one of the Cognitively Guided Instruction stories types presented in the article, Using Student Interviews to Guide Classroom Instruction: An Action Research Project (PDF). We also try connecting the stories to manipulatives (the blocks introduced at the beginning of the unit - see below) using the approaches presented in Teaching without Telling: Computational Fluency and Understanding through Invention (PDF) and Second Graders Cirumvent Addition and Subtraction Difficulties (NCTM). In the process, I explicitly point out my use of practitioner journals to inform my instruction.

Because not everyone needs support in this area, I am in the habit of posting on our class website PowerPoint Think Alouds of how I might go about solving some of the problems on this practice sheet.

Learners with access to the internet can download the Think Alouds and watch them at their convenience. This is how those first PowerPoints looked.

With current technological tools, like Jing, here is what I can post now.

These Think Alouds provide the learners who are struggling to apply the approaches they have read about a model of the approaches in action. It also models how teachers can use technology to offer different levels of support to learners. Some might call me the Wumanian Sal Khan, but I can assure you that these productions were not one-take affairs.

Do you have any Yis (Ones)?

Having successfully decoded the Wumanian number system (and further developed their understanding of place value), the preservice teachers begin to explore the concepts of composing and decomposing numbers. This is done through the use of centers - an instructional approach used successfully in many elementary classrooms. In this post, I share some of the more popular activities.

Yiwu Go Fish

As with many of the Wumanian center activities, this game has an Earth equivalent.

Summing Yiwu-yi Cubes

• With a partner, take turns rolling the dot cubes;
• Using symbols, write the amount shown on each individual cube (see example below);
• Write the total of the cubes combined (see example below); and
• Pass the cubes to your partner and repeat.

Collecting Coins

• Use a pencil and a paper clip to create a spinner out of the sectioned circle provided below;
• Take turns using the spinner to determine how many coins to add to your collection;
• Each turn, say how much you had, what you spun, and your new total after adding your spin.
• Keep track of the total worth of the collected coins using the rekenrek;
• Keep spinning until you reach a total of erwuwu pents; and
• If you finish early, play the game in reverse - giving the amount spun away until you reach na pents.

Fact Finding

On the addition table, identify which facts fit into which categories (note that some facts will need more than one color):

• Doubles
• Doubles plus or minus yi
• Count up yi
• Sums of yiwu
• Bounce off yiwu

Story Problems

In this center, learners work to solve a selection of Cognitively Guided Instruction "Joining" stories.

• There are si people in this group and si people in another group. How many rekenreks will I need if every person gets a rekenrek?
• I have er dot cubes. This center requires si dot cubes. How many more dot cubes do I need?
• A puzzle is made up of yiwu-yi pieces. Some of the pieces are already put together. There are san pieces left to complete the puzzle. How many pieces are already put together?

Yiwu Fingers

Yiwu is a benchmark number in the Wumanian system. Students who learn the different ways to decompose this benchmark are often more successful learning their facts and later on developing efficient mental strategies for larger numbers. This song and the accompanying motions (putting up the appropriate number of fingers and spinning around at "Now it's time for turn-arounds") is one way to support students in this area.

***

After the preservice teachers rotate through the centers, they are asked to reflect on the experience. I have them focus on how they might apply what they did in each activity to teaching composing and decomposing numbers in our system. Their responses form an assessment that allows me to evaluate whether or not they are able to transfer their experiences in base-yiwu to our base-tem system. Then I can then plan future learning activities based on their current level of understanding, not just what comes next in the curriculum.

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?