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.

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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.

Is mathematics about product or process?

Several years ago, a group of mathematics educators at GVSU decided to address an issue prevalent in our course, Mathematics for Elementary Teachers I. Many preservice elementary teachers in this course viewed mathematics from a product perspective. They considered only the answer as important. In response, we attempted to be more explicit regarding the mathematical processes by revising the course using the NCTM Process Standards as a framework. Below is one of the units that we developed around the theme of polygons.

In an effort to model promising teaching practices while maintaining the focus on the Process Standards, we introduced learning centers addressing Communication, Connections, Problem Solving, Representations, and Reasoning & Proof. The following centers came from a unit with Pentominoes as the theme. The preservice teachers rotated between these centers over the course of a two-hour class at the end of the unit.

Most of the ideas for the centers came from articles in Teaching Children Mathematics - NCTM's journal supporting improvement of pre-K-6 mathematics education. The content foci for the Pentominoes Unit were Measurement and Data Analysis and Probability. Some of the centers were closely related to the activities done during the unit, such as the Connection Center. Others, like the Reasoning & Proof Center, were new contexts that applied ideas found in previous lessons.

The preservice teachers looked forward to these centers at the end of each unit and were grateful for the instructional modeling of mathematics learning centers. Many had never experienced anything but traditional mathematics instruction in their K-12 experience. Still, I am not sure that one or two courses focusing on the Process Standards is enough to provide a more balanced perspective on what it means to do mathematics. After all, the process is as important as the product.