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Private: Learning Math: Number and Operations
Classroom Case Studies, K-2 Part B: Reasoning About Number and Operations (40 minutes)
In This Part: Exploring Standards
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. See Note 3 below.
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?
Principles and Standards for School Mathematics Copyright © 2000 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or redistributed electronically or in other formats without written permission from NCTM. standards.nctm.org
Standards are listed with the permission of the National Council of Teachers of Mathematics (NCTM). NCTM does not endorse the content or validity of these alignments.
Digi-Block® materials are used with permission of Digi-Block, Inc.
In This Part: Examining Children’s Reasoning
Here are scenarios from two different teachers’ classrooms, each involving young children’s developing ideas about number and operations. Snippets of students’ discussions are given for each scenario. For each student, consider the following:
• Understanding and Misunderstanding: What does the statement reveal about the student’s understanding or misunderstanding of number and operations ideas? Which ideas are embedded in the student’s observations?
• Next Instructional Moves: If you were the teacher, how would you respond to this student? What questions might you ask so that the student would ground his or her comments in the context? What further tasks and situations might you present for the student to investigate? See Note 4 below.
Problem B5
Second graders Leland, Randy, and Reed are given the following problem: Kent and his dog Nikki weigh 194 pounds when they are together on the scale. Kent weighs 146 pounds. How much does Nikki weigh?
Below is a snippet of their conversation:
Reed: Nikki weighs 48 pounds. I did it in my head. I first added 2 to 194 to get 196. Then I counted backward by tens until I got to 146 — 196, 186, 176, 166, 156, 146. That’s 5 tens, or 50. Then I subtracted the 2 to get 48.
Leland: I didn’t do it that way. I counted up by tens — 146, 156, 166, 176, 186, 196. That’s 50, and 2 less makes 48.
Randy: I wrote it down on a piece of paper, and 6 minus 4 is 2, and 9 minus 4 is 5, and 1 minus 1 is 0. So it’s 52.
a. What method did each student use to solve the problem? What does this tell you about how each student is thinking about the problem?
b. Do you think each student is ready to learn a new method? Why or why not?
Problem B6
Second graders Daniel, Tarra, and Mariko are given the following problem: Antonio has 35 marbles. Helen has 52 marbles. How many more marbles does Helen have than Antonio?
Below is a snippet of their conversation:
Mariko: They have 87 marbles all together.
Tarra: Helen has 35, 45, 55 — that’s 20. Count back 55, 54, 53, 52. She has 17 more.
Daniel: Fifty-two, 42, 32, then 33, 34, 35 — that’s 23 more.
a. What method did each student use to solve the problem? What does this tell you about how each student is thinking about the problem?
b. Do you think each student is ready to learn a new method? Why or why not?
Notes
Note 3
Digi-Blocks are manipulatives that reinforce the concept of grouping by tens. Students place small blocks into rectangular holders that fit exactly 10 blocks. The full holder then becomes a single entity — a group of ten. For more information on Digi-Blocks, go to http://www.digi-block.com/.
Note 4
You may wish to make a two-column chart, labeled “Understanding or Misunderstanding” and “Next Instructional Moves,” to help you organize your thinking for each problem. If you are working in a group, these charts could be the basis for a meaningful discussion on how to assess students’ understanding of the concept of subtraction and the processes for computation.
Solutions
Problem B1
a. The Digi-Blocks forced students to group in tens and hundreds. Most students started with the larger number of blocks. Students then took apart tens to obtain more ones, or hundreds to obtain more tens, as they completed the subtraction processes.
b. These blocks are automatically grouped in tens and hundreds. Students “see” the connections between the place value and the digits they are writing.
c. In this case, most students started with the larger collection and took away the smaller collection.
Problem B2
With this manipulative, the symbols students write on the paper match the actions they took with the blocks.
Problem B3
Ms. Weiss’s lesson is structured around understanding the meaning of operations, in particular, subtraction and addition. The students utilize a variety of methods and tools to solve problems. They also use multiple manipulatives, such as Digi-Blocks, which are particularly helpful in strengthening students’ ability to compute fluently and understand place value. The students using Digi-Blocks are also learning to count with understanding and recognize “how many” in sets of objects.
Solution B4
Ask students to talk about what they did with the blocks and what they wrote down on their papers. Listen for statements like this one: “I needed to take away one of the ones, and I didn’t have any ones. So I had to take apart one of the tens.” Students should be talking about taking apart and putting together units of ten.
Problem B5
a. Reed understands the concept and has good mental math skills, so he can do the computation in his head. He is well able to think of shortcuts to make mental computation easier.
Leland also has a good understanding of the concept and has the mental math skills to think of shortcuts for mental computation.
Randy does not understand the paper-pencil algorithm for subtraction with regrouping. He heard both Reed and Leland discussing their answers, but it didn’t seem to bother him that his answer didn’t match theirs.
b. Reed is probably ready to work on and understand other methods. For example, he could work on a written algorithm to match his informal understanding. We do not know how well he could do this with paper and pencil.
Leland is probably ready to work on and understand other methods. He also needs to work on a written algorithm to match his informal understanding. You can test his ability to do this type of computation with paper and pencil.
Randy is probably not ready to move on to new methods. He needs to work on some mental math skills and sense-making for subtraction with regrouping.
Problem B6
a. Mariko has entirely misunderstood the problem. She has found the sum of the two numbers rather than their difference. This may indicate that she is using key words without really reading the problem. She might have interpreted the key word “more” to mean addition.
Tarra counted up by tens, starting from 35, until she got close to the desired number, and then counted backward by ones until she reached the goal. She knew to count backward instead of forward here because she had passed the desired number, 52. This is sophisticated thinking. She appears to have a good understanding of the concepts and procedures.
Daniel counted backward by tens until he got close to the number, and then counted up by ones until he reached the desired goal. Daniel thinks that since he is now at 32, he must count up three to get to 35. He does not realize that he passed 35 on the way to 32.
b. Mariko is probably not ready to move on to new methods. She needs some hands-on practice with subtraction interpretations with smaller numbers. She does not appear to know that comparison requires subtraction. She should be discouraged from choosing an operation based solely on key words and encouraged to read and reread until the problem makes sense to her.
Tarra has a good understanding and is probably ready to move on to new methods. She now needs to be given more complicated problems, to test whether she can do subtraction using paper and pencil.
For Daniel, new methods may or may not be helpful. Although he understands the concept of subtraction, he still needs to practice with smaller numbers to strengthen his procedures for mental subtraction.
