392

Elementary Mathematics and Calculators: Lefs Think About It

Jane F. SchlClack Department of Mathematics

Clarence J. Dockweiler Department of Educational Curriculum and InstructionTexas A&M UniversityCollege Station, Texas 77843

Since the early 1970s, arguments have been developed bothfor and against the use of calculators in mathematics in theelementary grades (Burt, 1979). In an articlebasedon beginningexperiences with calculators in elementary classrooms. Bell(1979) identified somepedagogical, cumcular, and mechanicalconcerns. Even as Bell’s article was published, one of themechanical concerns, cost, was being addressed. What is ofinterest now is that two of the major concerns he identifiedagainst the use of the calculators in the elementary grades havevirtually disappeared: (a) solar cells haveprovided a simple andreliable source ofpower and (b) relatively inexpensive, durablecalculators arebeingproducedwith integer division and fractionnotation options for situations when these forms of responsesare more meaningful for students than are decimalrepresentations.

With the solution to these mainly mechanical difficulties athand, the state department of education in Texas, through theEESA- and Eisenhower-funded Texas Mathematics StaffDevelopment Project, has undertaken the task of addressingsome of the pedagogical and cumcular issues. Inserviceprograms developed for teachers ofkindergarten through grade6 present the use of the calculator as a tool for developingmathematics asproblem solving, communication, andreasoningwith emphasison themathematicalconnectionsbetween models,symbols, and ideas in the various content strands, as envisionedby the National Council of Teachers of Mathematics (NCTM)in their Curriculum and Evaluation Standards for SchoolMathematics (1989).

Although these inservice programs were developed prior tothe release of NCTM’s Professional Standards for TeachingMathematics (1991), the materials clearly attempt to addressthe concerns of the teaching environment, the development ofengaging tasks, and the discoursebetween students and teacher.In these inservice programs, teachers leam to integrate the useof the calculator into their mathematics instruction in order forstudents to generateinformation aboutagivenproblem, organizeinformation generated through the use ofthe calculator, explorepatterns in this information, makeconjectures aboutthepattems.and then use the calculator to aid in testing and modifying theirconjectures.

Concept Development in the Primary Grades

When asked to delineate personal concerns about the use of

calculators in the primary classroom, most teachers express adiscomfort with how the calculator will fit into the curriculumthey teach in mathematics. Is it developmentally appropriate?How will I find time to squeeze something else into a crowdedcurriculum? How does the use of the calculator relate to theemphasis on teaching with manipuladves? Isn’t it too abstract?And, of course, will it prevent students from learning the basicfacts?

This last concern is listed by teachers also as the main onethey expect parents and administrators to have about the use ofcalculators in the early grades. It is important that primaryteachers have theopportunity to identify theirpersonal concernsand what they perceive as others’ concerns about calculators inelementary mathematics classrooms-issues that they mustaddress in order to supportany decision to use the calculator forteaching mathematics. The following activities havebeen usedeffectively with primary teachers to alleviate some of theirinstructional concerns and to provide them with reasons forincluding the use of the calculator in their regular mathematicsinstruction.

Estimation, Number Sense, and Numeration

One of the goals for the use of the calculator in the primarygrades shouldbe its implementation as another typeofrecordingdevice. Young students should use the calculator as they wouldpaper and pencil, as a follow-up to active participation withmuch oral language involvement.

How Many? After listening to counters drop into a cup,students can press a number on the calculator to display thenumber of counters heard.

Counting Calculators. After working with concrete objectsto develop concepts ofvarious numerosities, students can leamto represent these amounts on the calculator by using theconstant function to count. By entering +1 on the calculator andthen pressing = for each of the counters as they are dropped, thestudents can generate the number representing the amount ofcounters in the cup.

Quick Counting. The calculator’s capability to displaynumbers can be connected directly to students’ concreteexperiences with thenumbersby using the calculator to reinforcethe patterns generated in base ten materials. Many youngstudents have difficulty counting with the combination ofhundreds, tens, and ones represented by the pieces in the base

School Science and Mathematics

Mathematics & Calculators393

ten materials. A calculator can be used to reinforce these Figure 2. Sample game of Make a Hundred.patterns. To count the hundreds squares, a student enters +100and presses = for each hundreds square that is picked up andplaced on a place-value board. As the student places thehundreds on the board, the display is read aloud, "100. 200,300....." When the hundreds are exhausted, the student enters+10 in the calculator and presses = for each tens piece, readingthe display as the pieces are placed on the board: "310. 320,330,...." The ones are included by entering +1 and pressing =for each unit while saying, "331, 332, 333,...."

Mixed Counting. As the students become more and morecomfortable with the place value patterns, they can begin toexperiment by entering the tens first, then the ones, then thehundreds, or various other combinations of the pieces. It willsoon become clear that a particular combination of place-valuepieces, no matter how they are counted, represents a uniqueamount. In thisway, thecalculatorprovides support forrecordingtheconnectionsbetween theconcretematerials and theirsymbolicrepresentation.

Whole Number Operations and Computation

The calculator can also be used to support the connectionsbetween concrete representations of whole number operationsand their symbolic representations.

Make a Hundred. In a game adapted from Family Math(Stenmark, Thompson, & Cossey, 1985), students work ingroups of three to try to generate a set of seven addends that willgive a sum as close to 100 as possible without going over 100.One student rolls the die and places the number rolled either inthe tens’ place or the ones’ place on a place-value chart (seeFigure 1).

Figure 1. Place-value chart for Make a Hundred.

Hundred Grid

Tens3

2

Ones080

Tens OnesCalculator Display:

58

One studentplaces the appropriate numberoftens orones onthe 10cmby 10 cm grid, and the third studentrecords thenumberon the calculator. After each of the next six rolls, the studentsdecide whether to place the number in the tens or ones column.place the appropriate place-value materials on the hundred grid,and add the appropriate number to the sum on the calculator tokeep a running total of the amount generated so far (see Figure2).

The number on the calculator display after each roll isrecordedalsoontheplacc"valuechartandrepresentedconcretelyon the hundred grid with place-value materials. By looking atthe place-valuepieces on the hundred grid, the students can seehow much their sum is in relation to 100 and how far they haveleft to go to get to 100. After each game, the students shoulddiscuss their strategies fordecidingwheretoplaceeachnumber.For example, how did they use the physical representation onthe hundred grid to help them make their decisions?

Volume 92(7), November 1992

Mathematics & Calculators

394

Application and Problem Solving

The previous activity can be extended by having studentsuse their calculator after each game to investigate optimumarrangements of their seven rolls. Did they place the numbersin the best possible positions as they were rolled in the game. orwould a different arrangement have given them a sum closer to100? Through logical analysis of the relative values in theplace-value chart, with the calculator providing computationalsupport for easy access to thechanges causedby rearranging thenumbers, students can be encouraged to use the informationgained from finding the optimum arrangements to improvetheir strategies while playing the game.

With these types of activities, where use of the calculator istied directly to the concrete representations ofthe numbers andoperations, teachers generate an environment where primarystudents can develop a healthy attitude toward the calculator asanother device that can be used to record the mathematicalsituations that they have actively investigated. With thisattitude toward the calculator as a tool that can be used torepresent mathematical situations, these students will be betterprepared to use the calculator appropriately in the intermediategrades for investigating patterns and testing conjectures.

Intermediate Grade Applications

Like teachers ofprimary grades, teachers of older children(grades 3 through 6) also express concern about the use ofcalculators. Their concerns differ somewhat because of thedifferent emphases at these levels. For example, they are stillconvinced that students will not leam their basic facts but theyalso wonder about a child’s ability to do pencil-and-papercomputation if the calculator is used extensively.A number of activities have been developed to illustrate the

effectiveness ofthe calculator in threebroad areas ofimportancein mathematics education. These three focal points for middlegrade use of calculators are: (a) concept development, (b)problem solving, and (c) reinforcement/maintenance. Thesethree areas were selected because they address the primaryconcern of learning at this level and because recent researchresults support calculator use in the attainment ofgoals in theseareas (Hembree & Dessart, 1986; Suydam, 1987).

Number and Numeration

Press On, Part I. This activity is a relatively simplecalculator activity with potentially big payoff. It is an attemptto getchildren carefully to identifyplace values as the calculatorportrays them. To begin, students clear the calculators. Foreach question posed, the learner looks at the calculator displayand responds to the questions (see Figure 3).

1. Press 1. What is the value of the number?2. Press 2. What happened to the I? What is the value of

the 2 in its new position? What is the total number?

3. Press 3. What happened to the I? What is the value ofthe 1 now? What happened to the 2? What is the value of the2? What is the total number?

4. Press 4. What happened to the I? What is the value ofthe I? What happened to the 2? What is the value of the 2?What happened to the 3? What is the value of the 3? What isthe total number?

Figure 3. Calculator displays/or Press On, Part /.

1. In12

3. 123

4. 1234

When entering whole numbers in the calculator, eachsuccessive digitcntered increases theplace valueofthe previousentry. Children’s correct oral responses give an indication oftheir understanding oftheplace valuerepresented. They shouldalso be able to generalize the result after observing thisphenomenon several times. The continuing development ofplace-valueconcepts is critical to a solid understanding ofideasassociated with number, and this activity provides a simple wayto extend these understandings.

PressOn,PartII. Thesecondpartoftheactivityextends theplace-value idea to decimal expressions. Thecalculatorhandlesdecimal entries in a different manner than it does whole numberentries (see Figure 4).

1. Clear the calculator and press the decimal point.2. Press 6. What is the value of the number?3. Press 7. What happened to the 6? What is the value of

the 6 now? What is the total number?4. Press 8. What happened to the 6? What is the value of

the6? What happened to the 7? What is the total value?5. Press 9. What happened to the 6? What is the value of

the 6? What happened to the 7? What is the value of the 7?What happened to the 8? What is the value of the 8? What isthe total value?

How do the successive entries in the two parts differ? Itappears that when whole numbers are entered, each successiveentry results in a place-value increase for the previous entry.Whereas, in decimal entries, after a digit is entered, its placevalue remains the same.

This type of activity permits the child to analyze carefullythe action of the calculator and to consider and identify placevaluesofnumbers displayedon thecalculator. Theidentificationof patterns and drawing conclusions suggests some higher-order thinking skills in the problem-solving area.

School Science and Mathematics

Mathematics & Calculators395

Figure 4. Calculator displays/or Press On, Part If.

1. I 0

2. .6

Figure 6. Proposed solutions/or Whole NumberGuesstimate.

a ai]x [5] Q

3.

4.

5.

.67

.678

.6789

2 2, 0 3 2

[H [HEBH]X

Theconceptual developmentaspects ofthis activity arequiteevident. The learner is expected to probe and discuss place-value ideas as they are reflected on the calculator. It is totallysymbolicandpresumesaconsiderableamountofpriorexperiencewith place values to allow the symbolic representation to bemeaningful. The ideas presented should contribute to thedevelopment of a mathematical power which the student willuse repeatedly in the development of related concepts, such asthe addition or subtraction algorithm.

Whole Number and Fraction Operations

Whole Number Guesstimate. An activity which is an oldfriend turns out to be an excellent activity which can be utilizedwith calculators. The problem is one in which the task is toconstructa wholenumber multiplication problem. The solver isto determine the maximum product when a three-digit numberis multiplied by a two-digit number (see Figure 5).

Figure 5. Template/or Whole Number Guesstimate.

Thedigitstobeutilizedarel,2,3,4,and5. They mustbe placedin the proper places to maximize the product. Successfulcompletion of this problem requires or, at least, encourages theproblem solver to try various combinations of digits. Thediscussion during a group solution results in a number ofcandidate solutions being proposed (see Figure 6).

2 2, 3 0 2

[H a Bx a (^2 2, 4 0 3

Some of the questions which should be asked are: Do wehave the largest product yet? What changes could be made toobtain a larger number? What convinces you that we do havethe largest number?

When agreement is reached that 22,412 is the largestproduct, a discussion regarding the cause is in order. Whatmakes it the largest? Why does the placement of the 4 and the5 in their proper positions result in the largest product?

Discussions like the preceding can lead the learner toconsider seriously some conceptual ideas relative to thedevelopment ofmultiplication, for example, the importance ofplace value and how place values are influenced in themultiplication algorithm. Many opportunities are provided tostrengthen the learner’s understanding of the multiplicationprocess. It is also fairly apparent that various problem-solvingstrategies areemployed in attacking this multiplication problem.Strategies such as guess-and-check are employed repeatedly intrying various combinations ofdigits to maximize the product.The repeated use of multiplication facts, although calculatedon the calculator, will certainly give the learner frequentexposure to these important bits ofinformation. That exposurewill contribute to the familiarization which will reinforce thefacts and permit a level of maintenance for other applications.In this one activity, therefore, all three of the identified foci ofcalculator use-concept development, problem solving, andreinforcement/maintenance� areemployed orbecome evident.

Volume 92(7), November 1992

Mathematics & Calculators396

Extensions of Whole Number Guesstimate. If furtherextensions are desired, consideration could be given to othersets offive digits. What if the set includes a repeated digit (e.g.,1.2.3.4,4)? What if the set includes a zero (e.g., 1,2.3.4,0)?Are theresults similar? Aretheproductspredictable in the samewayas theexample? Although theproblem could beconsidereda puzzle and a fun activity, a proper discussion and leadingquestions can provide a marvelous opportunity to buildmathematical concepts.

Fraction Guesstimate. Adifferentapproach to guessingandchecking extends theprevious activity to fraction considerations.The availability of calculators that permit the symbolicmanipulation offractions allows learners to look at the additionof fractions in this way.

Students consider a set offour numerals (e.g., 2,3,4,5) andwhere they should be placed in the fraction addition grid toproduce the largest sum (see Figure 7).

Figure 7. Template and examples for Fraction Guesstimate.

Di

DElITH]-�. 4.

B

D���� ==

DBifH]-�� =

H]

22���

15

22���.

8

Frequently, a first try to create a largest sum results in thesecond example in Figure 7. There is often a reluctance tocreate a fraction greater than one. Presumably, this reluctancemay be because of the improper nature of such fractions. Thisresult only serves to illustrate the conceptual nature of theactivity and the type of discussion which could be generatedvery early.A second try atobtaining the greatest sum might result in the

third example in Figure 7. Why not try a descending orderplacement? How does that influence the sum? Could it be stilllarger? Where should the largest numeral be placed? Whatabout the smallest?

Extensions of Fraction Guesstimate. It is probable that,from theprevious experiencewith whole numbermultiplication,a pattern of digit placement evolved to obtain the largestproduct. Is it possible that a similar pattern could exist for thiscase in which the largest fraction sum is to be determined?Other questions that might lead to further conceptualdevelopment and understanding are: Would a different set of

four digits require a different pattern? Could a zero be includedin the set of four digits? What if one of the digits was repeatedin the set (e.g.. 3,4,4,6)? The ease of symbol manipulationprovided by the calculator permits the learner to look carefullyat some very important mathematical concepts without theconcern for obtaining the correct answer through paper andpencil computation.

Numeration and Probability

Roll Along. This activity is one in which the full displayrange of the calculator may be utilized. By means of a dice ornumber cube, the players are to fill in an 8-digit number (seeFigure 8).

Figure 8. Templates for Roll Along.

Player 1

Player 2

The rules are relatively simple:1. Each player, in turn, rolls a die and places the obtained

number in one of the eight places available and enters theappropriate value in the calculator.

2. After a place-value position has been filled, it may notbechanged.

3. The winner for each round is the player who obtains thegreatest 8-digit number.

For those who are concerned about a game which might betoo competitive, Roll Along might be an answer since the digitsobtained are dependenton thedie and notthe player. All playersshould have equal opportunity to win. Of course, strategiesshould be developed to place the digits most advantageously.Higher-order thinking skills are utilized in determining theappropriate place for each digit as it is obtained.

In addition to theproblem-solvingcomponentjust described,considerable intuitiveconceptdevelopmentin probabilityshouldbe taking place. Using a typical die, how many digit choices arethere for each turn? If three 2s are obtained during a player’sfirst three turns, are they likely to obtain another on the nextroll? If a 5 is obtained on one of the early turns, where shouldit be placed, or what is the likelihood of obtaining a 6 later?

When entering the digit in the appropriate place in thecalculator, what place-value skills are being used or extended?How does the activity of entering the digit in the calculatordiffer from simply writing the digit in the proper place on thepaper record? Does one require a greater understanding of therelated place-value concepts?

This activity is another relatively simple example whichtakes advantage of the calculator’s capabilities and permits thelearner to experience many different mathematical ideas. Allthreeofthe identifiedcmphases-conccptdevelopment,problem

School Science and Mathematics

Mathematics & Calculators397

solving, and reinforcement/maintenance-are included in theactivity’s completion and related discussion.

Modifications of Roll Along. Various modifications can bemade which increase the value of the simple activity.

1. For younger learners, it may be advisable to restrict theplace values to three or four to stay within reach oftheir abilities.It is an equally effective activity and merits an early experienceto the related ideas.

2. It may also be worthwhile to ask each child to state orallythe value of the digit that is being entered for each turn. Theconnection of the correct terminology to the concept is anextremely important ability to develop in the process of eachlearner internalizing the complexities of a place-value system.

3. An interesting extension is to have each player look attheir collection of digits at the completion ofone round to see ifa rearrangement of the digits could produce a larger number.

4. Would the results be different if a different number cubewould be used? What ifa number is repeated in the cube so thatthe possible digits might be 1,2.3.3,4.5? Would the results bedifferent?

An Example of Implementation

Most teachers who have experienced these inserviceprograms return to the classroom eager to incorporate the use ofthe calculator in their mathematics instruction. Onesuch teacherbegins using the calculator with a small group in order to buildher confidence in working with this new teaching device. Shehas identified what she hopes will be an intellectually engagingstudent activity to strengthen the place-value concepts theycurrently have been studying. �

Mrs. S. has gathered six other fourth-grade students arounda table at the side of theroom. Each student has a calculator andis preparing for the newactivity the teacherhas announced. Mrs.S. records a telephone number on the chalkboard

3234045and says, "I would like to display this phone number on mycalculator. But I can use only the 0 and 1 number keys, theoperation keys for addition, subtraction, multiplication, anddivision, and the = key. All of the other keys are broken. Howam I going to get the display I want? Work with your partner fora few minutes to see what ideas you come up with."

After a few minutes. Mrs. S. calls for the students’ attentionand asks them to share their ideas. Terry and Jean say that theytried 1+1+1+1+1+... and they feel that they wouldeventually get to the number that she wanted to display. Otherstudents are nodding their heads. Alexis and Steve suggest thatusing the 0 to add 10 at a time or 100 at a time would be faster.In a flurry ofdiscussion, all the students begin punchingbuttonson their calculators.

Mrs. S. allows them to explore for a few more seconds, thenregains their attention and says, "If this phone number were aregular number, what would each of these digits represent?"

The students respond together as Mrs. S. points to eachdigit. "The 5 is five ones, the 4 is four tens, the 0 is zerohundreds, the4 is four thousands, the 3 is three ten-thousands,the 2 is two hundred-thousands, the 3 is three millions."

"Now," says Mrs. S.. "does that give you any ideas abouthow you could use Alexis’ and Steve’s technique for adding 10or 100 at a time to reach the number faster? See ifyou can findthe quickest way to display this number using only the 0. 1,operation, and = keys."

When all the pairs of students have had a chance to form aprocedure and test it, Mrs. S. calls them back together for somediscussion. Each group is allowed to explain what theyconsider to be the quickest way to display the given number.One group explains that they added 1,000,000 three times,100.000 two times, 10,000 three times, 1000four times, 10 fourtimes and 1 five times. Alexis and Steve argue that their wayis quicker because they have combined some of the numbersinto addends like 1011. The third groups says that they mixedsome Os and Is in their addends, but did not use 1011 likeAlexis and Steve did. Mrs. S. asks all thegroups, "Do you thinkwe have found the quickest way yet? Maybe if we organizedour ideas, we would be able to tell for sure. Let’s use your ideasabout place value and show them in a chart. Howmany did wesay were in the ones’place? Five. So wewill need to add at leastfive ones." She looks around to make sure thateveryone agreeswith that statement. Since they have already tried differentways of reproducing the number, they all feel certain that thisstatement is true.

Mrs. S. records five Is under the 5 on the chart and says"How many tens do we need? Four. Do we need to list themseparately or can we combine them somehow with the digitsalready in the ones’ column tliat wo know we need?" Some ofthe students respond excitedly that the five ones and four tenscouldbecome four lls and a 1. Alexis and Stevecomment thatthat is the way theydecided was the quickest. Mrs. S. continueswith this organization by asking students to help her place theappropriate number of 1 s in each of the place-value columns.As they go along, they discuss the need for a 0 wherever thereis a space between the Is and whether or not there is a need for"0"s in every space that is empty on the chart. At the end, theynotice that there are only Os in the hundreds’ column, sincethere was a 0 in the hundreds’ placeofthe original number, andone 0 under the 2 (see Figure 9).

After testing theaddends generated with this organizationalprocedure, the students are convinced that they have found thequickest way to display the given telephone number with thegiven restrictions. The students then are given the task to findthe quickest way to display their own phone numbers (or onethey have chosen from a telephone book) and use a chart orother explanation to verify that it is the quickest way. Thesestudents’ level of interest and success encourage this teacher tolook for more ways to include the use of the calculator in everystudents’ mathematics learning experiences.

Volume 92(7), November 1992

Mathematics & Calculators398

Figure 9. Charts for the analysis of the quickest sum.

2.2.3.^0.4.5.11111

2. 2. 3.

3-2. 3.1 1 11 1 11 0 1

–11111

0.

0.0000

4. 5.1 11 11 11 1

1

4. 5.1 11 11 11 1

1

Conclusion

The calculator provides unique opportunities for younglearners to develop ideas and investigate relationships. Manyprojects in this country and in other countries, such as thePRIME project in England (1988), provide new insights andactivities that promote the message of the NCTM curriculumstandards. It is no longer a question of whether the calculatorshould be used, but how and when it can best be used andintegrated into thecurriculum at all grades, kindergarten throughgrade 8.

References

Bell, M. S. (1979). Calculators in elementary schools? Sometentative guidelines and questions based on classroomexperience. In B. C. Burt (^^.Calculators: Readingsfromthe Arithmetic Teacher and the Mathematics Teacher (pp.55-62). Reston, VA: National Council of Teachers ofMathematics.

Bun. B. C. (Ed.). (1979). Calculators: Readings from theArithmetic Teacher and the Mathematics Teacher. Reston,VA: National Council of Teachers of Mathematics.

Hembree, R.. & Dessart, D. J. (1986). Effects of hand-held

calculators in precollege mathematics education: A meta-analysis. Journalfor Research in Mathematics Education,17(2), 83-99.

National Council of Teachers of Mathematics. (1991).Professional standards for teaching mathematics. Reston,VA: Author.

National Council of Teachers of Mathematics. (1989).Curriculum and evaluation standards for schoolmathematics. Reston, VA: Author.

PRIME Newsletter. (1988). Cambridge, England: PrimaryInitiatives in Mathematics Education Project.

Stenmark, J. K., Thompson, V.. & Cossey, R. (1986). Familymath. Berkeley: Regents, University of California.

Suydam, M. N. (1987). Research on instruction in elementaryschoolmathematics: A letter to teachers. Columbus, OH:ERIC Clearinghouse for Science, Mathematics, andEnvironmental Education.

Course syllabianddescriptions available

from the ...

NSFModel Middle School Mathematicsand Science Teacher PreparationProgram at Potsdam College

The syllabi are on a Macintosh� compatibledisk in both Microsoft fi Word and text files. Atpresent, they can be obtained free of charge.

Forinformationregarding theinterdisciplinaryscience, mathematics, and education coursesdeveloped forthemodelprogram and/orto obtainthe disk contact:

Dr. Timothy J. Schwob, DirectorNSF Model Program308 Satterlee HallPotsdam College

Potsdam. NY 13676(315)267-2474

schwobtj@SNYPOTVX (bitnet)

School Science and Mathematics