Session 10, Part B:
Reasoning About Number and Operations (40 minutes)

In This Part: Exploring Standards | Examining Children's Reasoning

The National Council of Teachers of Mathematics has identified number and operations as a strand in its Principles and Standards for School Mathematics. In grades pre-K through 12, instructional programs should enable all students to do the following:

 • Understand numbers, ways of representing numbers, relationships among numbers, and number systems • Understand the meaning of operations and how they relate to one another • Compute fluently and make reasonable estimates

In pre-K through grade 2 classrooms, students are expected to do the following:

 • Understand various meanings of addition and subtraction of whole numbers and the relationship between the two operations • Develop and use strategies for whole-number computations, with a focus on addition and subtraction • Develop fluency with basic number combinations for addition and subtraction • Use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and pencil, and calculators • Count with understanding, and recognize "how many" are in sets of objects • Use multiple models to develop initial understandings of place value and the base ten number system • Connect number words and numerals to the quantities they represent using various physical models and representations

The NCTM Number and Operations Standards state that students should "develop a solid understanding of the base-ten numeration system and place-value concepts by the end of grade 2... Using concrete materials can help students learn to group and ungroup by tens. For example, such materials can help students express '23' as 23 ones (units), 1 ten and 13 ones, or 2 tens and 3 ones. Of course, students should also note the ways in which using concrete materials to represent a number differs from using conventional notation. For example, when the numeral for the collection [23] is written, the arrangement of digits matters -- the digit for the tens must be written to the left of the digit for the units. In contrast, when base-ten blocks or connecting multi-cubes are used, the value is not affected by the arrangement of the blocks" (NCTM, 2000, p. 81).

As you watch another video segment from Ms. Weiss's class, think about how the students are developing this understanding of number and operations.

 Video Segment In this video segment, two groups of students use Digi-Blocks to solve subtraction problems. Note 3 If you are using a VCR, you can find this segment on the session video approximately 16 minutes and 22 seconds after the Annenberg Media logo.

Problem B1

 a. How did the students use the Digi-Blocks to represent the problem? b. What processes did the students use to group the Digi-Blocks? c. What subtraction strategies did the students consider?

 Problem B2 How did the Digi-Blocks help students relate their actions to the written algorithm?

 Problem B3 What are some ways that you see the NCTM Standards being incorporated into Ms. Weiss's lesson?

 Problem B4 Embedded in the children's explanations of solving the subtraction problems are early understandings of place value. How could you extend this conversation to formalize these notions?