9 781292 042053

ISBN 978-1-29204-205-3

Learning Mathematics in Elementaryand Middle Schools

A Learner-Centered ApproachCathcart et al. Fifth Edition

Learning Mathem

atics in Elementary and M

iddle Schools 5

e

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Using number and language cards in the classroom canhelp make the transition from horizontal to vertical notationmore natural. Sentences such as those shown in Figure 7-18can be arranged in vertical format. Initially, the sentencesmight be formed as a simple 90-degree rotation of thehorizontal sentence (Figure 7-19). Later, the format can

be altered to conform more closely to conventional nota-tion. See Figure 7-20.

Over a period of time, introduce the conventional sym-bols for the different action words. Also introduce the “=”symbol for words such as “makes” and “leaves.” Write theseconventional symbols on cards as well and have children usethem to generate sentences for problems and models in thesame way they did with the natural language cards.

Children should understand that the “=” sign means “isthe same number as” or “is another name for.” Havingchildren write several expressions equivalent to a givenexpression should facilitate this understanding. For exam-ple, give children ___ and ask them to write atleast three true expressions (not just one number) in theblank. They may respond with:

Note that the notation for division is particularlyconfusing to children. Three symbolic representations arecommonly used for division: and or . Thelatter format usually is delayed until children are studyingfractions. The first usually is read as “six divided by two”and the second as “two goes into six,” although it shouldalso be read as “six divided by two.” The second repre-sentation is the most common, but also it is the only sym-bolic representation that should not be read from left toright, as “two goes into six.” It is important to help chil-dren connect the phrase “six divided by two” to each ofthe common symbolic forms.

ConclusionIt is critical that children build a sound understanding ofwhole-number operations. Therefore, children should beallowed time to manipulate and solidify their ideas. Solv-ing word problems of many different types is necessaryfor children to develop this “operation sense.” To rush onto “more advanced” work is a mistake that often comesback to haunt children and teachers.

626/22�66 , 2,

5 + 7 = 24 , 2 5 + 7 = 15 - 3

5 + 7 = 6 + 6 5 + 7 = 10 + 2

5 + 7 =

Developing Whole-Number Operations: Meaning of Operations

add plus makes

cross out equal s

bags of joined by

shared by take away

FIGURE 7-17 Phrases Used in Word Problems

is

leaves

makes

cross out

equal s bags of

joined by

shared by

8 4 12

3 5 15

7 2 5

21 3 7

FIGURE 7-18 Sentences Representing Word Problems

makes

joined by

8

4

12

FIGURE 7-19 A Vertical Sentence for a Word Problem

makes

joined by

8

4

12

FIGURE 7-20 Sentence in ConventionalPosition

130

Explore: In pairs or small groups, have students solve asimilar problem and draw a picture of their solution.

Summarize: Coming together as a whole class, selectone pair of students to come to the front of theroom and explain how they solved the problemand to show the picture of their solution. Focus ontheir explanations of how they determined howmuch larger one set was than the other.

FOLLOW-UP

Complete the following questions.

1. Consider any cautions regarding selecting themanipulatives to use in this lesson. Would othermanipulatives work well? Might you want toavoid using certain manipulatives or objects?Why? What would be the advantages and/or dis-advantages of each manipulative?

2. Two types of number sentences are appropriatefor comparison problems: 4 + __ = 9 and 9 – 4 = __. How might you help children learnhow to write a number sentence for comparisonproblems?

3. What might the next lesson focus on, and why?

Grade level: First

Materials:

• At least one manipulative, such as Unifix cubes,or assorted counters or objects

Lesson objective: Students will use objects to cor-rectly represent and compare whole numbers.

Standards link: Students use a variety of models tomodel “part-whole,” “adding to,” “taking awayfrom,” and “comparing” situations to develop anunderstanding of the meanings of addition andsubtraction and strategies to solve such arithmeticproblems (NCTM (2006) Curriculum Focal Points,Grade One).

Launch: Pose the following problem:

Sharon has 9 marbles, and Tom has 4 marbles.How many more marbles does Sharon havethan Tom does?

Ask the students how they might solve the problem.Solicit recommendations about how to solve theproblem, including modeling each quantity withcounters, making a one-to-one correspondence be-tween objects in both sets, and counting how manymore are in the larger (Sharon’s) set.

Sample Lesson

A Lesson on Solving Comparison Problems

IN PRACTICEComplete the following activities to include in your pro-fessional portfolio.

1. Interview a child to assess his or her understanding ofoperations. (You may choose to use problems fromthe Early Number or Place-Value Interviews on the

IMAP clips or problems that you write.) Describe theproblems you asked the child to solve and the strate-gies he or she used to solve problems. Describe whatthe child seemed to understand and what understand-ing the child still needs to develop.

2. Write a lesson plan to help introduce children to theJoin type of addition problems.

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Sausalito, CA: Math Solutions.Creative Publications. (1994). The Maharajas’ Tasks: Investi-

gating Division. Mountain View, CA: Creative.National Council of Teachers of Mathematics (2006).

Curriculum Focal Points for Prekindergarten through Grade8 Mathematics. Reston, VA: Author.

Ohanian, S., & Burns, M. (1995). Math by All Means: Division,Grades 3–4. Sausalito, CA: Math Solutions.

Richardson, K. (1999). Developing Number Concepts, Book 2:Addition and Subtraction. White Plains, NY: Seymour.

Richardson, K. (1999). Developing Number Concepts, Book 3:Place Value, Multiplication, and Division. White Plains, NY:Seymour.

Ward, S. (1995). Constructing Ideas about Number Combina-tions. Mountain View, CA: Creative.

Children s LiteratureAddition and SubtractionBecker, John (2007). Seven Little Rabbits. New York: Walker

Books for Young Readers.Derubertis, Barbara (2005). Count On Pablo. New York: The

Kane Press.Jonas, Ann. (1997). Splash. New York: HarperTrophy.Merriam, Eve (1996). Twelve Ways to Get to 11. New York:

Aladdin.Murphy, S. (1996). Too Many Kangaroo Things to Do! New

York: Harper Trophy.

Multiplication and DivisionGiganti, Paul (1999). Each Orange Had 8 Slices. New York:

HarperTrophy.Hulme, Joy N. (1999). Sea Squares. New York: Hyperion.Mahy, Margaret (1993). 17 Kings and 42 Elephants. New York:

Puffin.Neushwander, Cindy (1999). Amanda Bean’s Amazing Dream, a

Mathematical Story. New York: Scholastic Press.Pinczes, Elinor J. (1995). A Remainder of One. Boston:

Houghton Mifflin.Pinczes, Elinor J. (1999). 100 Hungry Ants. Boston: Houghton

Mifflin.

3. Write a lesson plan to help introduce children to theEqual Groups type of multiplication and divisionproblems.

Now go to Topic 6: “WholeNumber Operations” in the

MyEducationLab (www.myeducationlab.com) for yourcourse, where you can:• Find learning outcomes for “Whole Number Opera-

tions” along with the national standards that connectto these outcomes.

• Apply and practice your understanding of the coreteaching skills identified in the chapter with a Build-ing Teaching Skills and Dispositions learning unit.

• Complete Assignments and Activities that can help youunderstand the chapter content more deeply.

• Complete enVision MATH Sample Curricula assign-ments that allow you to examine and work withchapters from enVision MATH, a K-6 mathematicsprogram.

• Check your comprehension on the content coveredin the chapter by going to the Study Plan in the Book-Specific Resources for your text. Here you will beable to take a chapter quiz, receive feedback on youranswers, and then access Review, Practice, and Enrichment activities to enhance your understandingof chapter content.

• Go to the Book-Specific Resources for Chapter 7 toexplore mathematical reasoning related to chaptercontent in the Activities section.

LINKS TO THE INTERNETProTeacherhttp://www.proteacher.com/100000.shtmlContains links to lessons to help children understand andpractice addition, subtraction, multiplication and division.

RESOURCES FOR TEACHERSReference Books: Whole-Number OperationsBrodie, J. (1995). Constructing Ideas about Multiplication and

Division, Grades 3–6. Mountain View, CA: Creative.Burns, M. (1991). Math by All Means: Multiplication, Grade 3.

132

NCTM

Developing Your Math Teaching SkillsWhen you have finished study-ing this chapter, you should beable to do the following:

• Name the three componentsof instruction on basic facts.

• For each whole-number op-eration, describe some think-ing strategies that childrencan use. Describe severalthinking strategies for eachoperation.

• Explain the role of consoli-dating activities for drill andpractice. Describe several ofthese activities.

• Discuss how games are use-ful in promoting the immedi-ate recall of basic facts.

Developing Whole-Number

Operations: Mastering

the Basic Facts

CONNECTING WITH THE STANDARDS

The National Council of Teachers of Mathematics (NCTM) Principlesand Standards (2000) affirms the importance of children’s developingproficiency with basic facts and algorithms but cautions against overem-phasizing the memorization of facts before understanding is developed,or to the exclusion of other important topics.

The NCTM (2006) Curriculum Focal Points discusses learning basicfacts in grades one through four. They state that first-grade students “useproperties of addition (commutativity and associativity) to add wholenumbers, and they create and use increasingly sophisticated strategiesbased on these properties (e.g., “making tens”) to solve addition andsubtraction problems involving basic facts” (NCTM, 2006, p. 13), whilesecond graders “use their understanding of addition to develop quickrecall of basic addition facts and related subtraction facts” (p. 14).Similarly, third-grade students “use properties of addition and multipli-cation (e.g., commutativity, associativity, and the distributive property)to multiply whole numbers and apply increasingly sophisticated strate-gies based on these properties to solve multiplication and division prob-lems involving basic facts” (p. 15), while fourth-grade students “useunderstandings of multiplication to develop quick recall of the basicmultiplication facts and related division facts” (p. 16).

In learning the basic facts, children focus less on real-life problemsand more on the relationship between models and the symbolic repre-sentation of the facts. Children need many opportunities to discuss thischange in emphasis and to translate from physical model to symboland vice versa before they can be expected to operate solely at thesymbolic level.

ASSESSING MATHEMATICS UNDERSTANDING

In order to become familiar with the mathematics concepts andprocedures discussed in this chapter, take a few minutes and com-plete the following Preassessment. Answer each question on your own and think about how you got the answer. Then think about how elementary-school children think about each problem

From Chapter 8 of Learning Mathematics in Elementary and Middle Schools: A Learner-Centered Approach, 5/e. W. George Cathcart. Yvonne M. Pothier. James H. Vance. Nadine S. Bezuk. Copyright © 2011 by Pearson Education. All rights reserved.

© 2005 LessonLab, a division of Pearson Education, Inc. All rights reserved.

133

She then rotates the paper 180 degrees, askingstudents what number sentence it now represents(3 + 1 = 4).

and any possible misunderstandings some childrenmight have about each problem. If you are able, ad-minister this assessment to a child, and analyze hisor her understanding of these topics.

Part 1: Addition Facts

1. 8 + 0 = ___ 2. 7 + 1 = ___

3. 4 + 2 = ___ 4. 3 + 5 = ___

5. 5 + 5 = ___ 6. 9 + 2 = ___

7. 6 + 4 = ___ 8. 8 + 6 = ___

Part 2: Multiplication Facts

9. 6 * 0 = ___ 10. 8 * 1 = ___

11. 4 * 2 = ___ 12. 3 * 5 = ___

13. 7 * 4 = ___ 14. 8 * 6 = ___

15. 7 * 6 = ___ 16. 8 * 7 = ___

Examining Classroom Practice

Teacher Alma Wright uses dominoes to help her first-and second-graders identify different combinationsof numbers for a given sum (Annenberg Video Series).The children gather in a circle around a large set ofdominoes arranged on the floor. Ms. Wright asks thechildren to find dominoes that show 4. One childchooses a domino with 4 dots on one end and 0 dotson the other. Another child chooses a domino with 1 dot on one end and 3 dots on the other end. Ms. Wright helps the students verbalize the numbersentence this domino represents by saying, “Oneplus three equals four.”

After children identify other dominoes that show4 and state the corresponding number sentence,Ms. Wright uses stick-on dots to create a pictorialrepresentation of the first domino selected, show-ing 0 + 4 = 4.

Ms. Wright then encourages children to work insmall groups to find dominoes that have a sum of five:first locating the dominoes, then using stick-on dotsto create a picture of each domino, and finally writ-ing number sentences to represent each domino theychoose. For groups that finish quickly, Ms. Wrightposes the task of finding three dominoes that add toa total of eight.

As the children solve problems, Ms. Wright cir-culates around the room, asking clarifying questionsand posing follow-up tasks as needed. Most studentscount the dots to determine the total, while other stu-dents seem to have memorized a few of the numbersentences.

Analyzing Assessment Results

Analyze each problem in the Preassessment. Howdid you (and/or the student you administered the assessment to) solve each of the problems? Whichfacts were harder? Why do you think that was?

1. 8 + 0 = ___

2. 7 + 1 = ___

Discussion: These problems are two of the easi-est addition facts. Both involve adding 0 or 1. Theseare often some of the first addition facts that childrenmemorize.

3. 4 + 2 = ___

4. 3 + 5 = ___

Discussion: These problems are a bit harder, butstill fairly easy. Both involve adding 2 or 3. Countingon still works easily for these facts. Children memo-rize these facts fairly easily.

5. 5 + 5 = ___

Discussion: This problem is a “double,” mean-ing that both addends are the same. Doubles areoften some of the first addition facts that childrenmemorize.

6. 9 + 2 = ___

Discussion: This addition fact involves 9. Chil-dren can reason that adding 9 is the same as adding 10

Go to the Video Examplessection of Topic 6: “Whole

Number Operations” in the MyEducationLab for yourcourse to view the Annenberg clip “Domino Math.”

As children watch, she moves 1 dot from oneend of the domino picture to the other, creating thenumber sentence 1 + 3 = 4.

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