Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

## Place-Value Centers

The goals of the NCTM's reasoning process standard are that "in grades K-4, the study of mathematics should emphasize reasoning so that students can-

• draw logical conclusions about mathematics;
• use models, known facts, properties, and relationships to explain their thinking;
• justify their answers and solution processes;
• use patterns and relationships to analyze mathematical situations;
• believe that mathematics makes sense."
(NCTM, Curriculum and Evaluation Standards for School Mathematics, p. 29) Video Overview
Before this lesson, students worked on adding and subtracting single digits, counting to 100, measuring, exploring Unifix cubes, and putting objects into groups of ten. Students are now presented with a calendar mathematics activity. They review how to display quantities on a place-value chart and how to record amounts with numerals. Students are divided into groups to work at centers with activities that reinforce place-value understanding. The centers are (1) measuring items with Unifix cubes, (2) counting beans and stones, (3) representing numbers from a hundreds chart with base-ten blocks, and (4) racing to make the longest train with Unifix cubes, then counting these cubes by tens. Students develop an understanding of the numeration system by relating counting, grouping, and place-value concepts. Some students use ungrouped materials to form groups of ten and then to count by tens and ones. Other students use pregrouped place-value materials to represent specific numbers. Throughout the lesson, place-value language, such as saying the number of tens and ones, 1 ten and 3 ones, is related to the standard oral name, thirteen, and to the standard symbolic representation, 13. Students also measure items, such as their heights, by linking Unifix cubes and then breaking the cubes into tens and ones for counting to find the total height. After each group completes three centers, the class reconvenes to discuss its accomplishments and discoveries.

Topics for Discussion
For Teacher workshops

The following areas provide a focus for discussion after you view the video. You may want to customize these areas or focus on your own discussion ideas.

Developing Place-Value Concepts

1. How did the calendar activity help students develop an understanding of place value? Describe the long-term nature of the calendar activity and the benefits it has for students' learning.

2. Specify the four learning centers used in this lesson and identify the mathematical content and connections in each center.

3. The informal place-value language used by teachers varies. What language did Ms. Vigstrom use? What other words describe place values? Define differences in how children might hear these words.

4. An important development in students' place-value understanding is their move beyond counting by ones to counting by tens and ones. Cite observations of this progression in the video.

5. Describe ways to help students integrate their ability to count the number of tens, such as 1 ten, 2 tens, 3 tens, with counting by tens, such as ten, twenty, thirty, and so on.

6. What other place-value activities could be planned for students?

Analyzing Place-Value Materials

1. 1 Ms. Vigstrom set up several place-value centers with different materials. Identify the various manipulatives used in this lesson. How effective were the materials in helping develop place-value concepts? Which materials were most effective and why?

2. Both ungrouped materials (Unifix cubes, beans, buttons, and stones) and pregrouped materials (base-ten blocks) were used for manipulatives in this lesson. Describe the advantages and disadvantages of each type of material. What is the developmental progression in using these materials?

3. The previous materials are considered proportional place-value materials. Nonproportional place-value materials and activities include money-such as trading a penny for a dime-and poker chips-such as trading a blue chip, which has a value of 10, for a white chip, which has a value of 1. What are the advantages and disadvantages of proportional and nonproportional materials for developing place-value concepts?

4. The groups of children rotated among the learning centers during the lesson. Each group was able to work at three different centers. Identify the pros and cons of students' using different manipulatives in the same lesson. Why would a teacher want students to use a variety of materials that represent the same mathematical idea?

Extension
Assessment of Place-Value Knowledge

Develop assessment strategies for place value. These strategies may include performance tasks, interviews, or observational checklists. For each strategy, clearly identify the criteria for judging students' understanding.

Learning Centers

Define additional centers for developing place-value concepts. List various types of materials to be used in the centers and ways to obtain those materials. Develop management techniques for introducing students to the centers and for using the learning centers.