Mathematics for Elementary and Middle School Gifted Students

576

Mathematics forElementary and Middle SchoolGifted Students

Ray Kurtz

Meeting the mathematical needsof the gifted, creative, and talented(GCT) students is finally in vogue.Federal and state mandates have en-couraged local districts to developan educational program for thesestudents. The form this programtakes is almost as variable as thenumber of programs coming intoexistence. Much is being writtenabout the general characteristics ofthis group of students. However,less has been written which specif-ically addresses how to challengethese students with mathematics.Probably the greatest fallacy that

exists is that gifted, creative, or tal-ented students have the self disci-pline to be directly channeled intoindependent study. This is mani-fested in the classroom by teacherswho, by their actions, are content tolet the capable students more or lessforage for themselves. The oldcliche that "the excellent student

School Science and MathematicsVolume 83 (7) November 1983

Mathematics/or Gifted Students 577

will learn in spite of the teacher" is practiced far more than we would liketo admit. A statement that is commonly heard which over stresses inde-pendent study is "challenge the gifted; bring the gifted and bookstogether." The implication is that the gifted will independently grab thebook and pursue advanced knowledge. Even though this works in manycases, it fails in far too many others. The failure of present practices aresummarized in the research which shows that over 17 percent of giftedstudents are high school dropouts (Maryland, 1972).

Grouping Students

Gifted, creative, and talented students need the social and emotional sup-port of their true peers (intellectual and chronological), in order to prog-ress and maintain their positive self concept. These students also need theencouragement that group interaction of able minds can provide. Thesestudents must be grouped for interaction if there is to be enrichment andnot just rote learning; otherwise we face the problem of a single GCT stu-dent isolated in a class where excelling beyond the class norm can be verydifficult�for both students and teacher.

^The old cliche that the excellent student will learn in spiteof the teacher is practiced far more than we would like to

admit/9______________________________

Organizing for enrichment

If the enrichment is to be provided in the regular classroom, UnderhilPs1981 model may be used. The model provides for enrichment as often asa student demonstrates readiness for it. If there are not enough able stu-dents available in one classroom, a plan for clustering students from twoor more classes should be explored. It has been observed that it is com-mon for the regular classroom teacher to feel guilty planning and prepar-ing challenging activities for GCT students. This is especially true whenthe other students ask why they don’t get to do the "fun things"provided for the GCT students. If this happens, it may be that the activi-ties are not challenging. Basic drill packaged in rows is never consideredenrichment. However, basic drill packaged into puzzles, mazes, etc. toooften comprises enrichment programs. Even the materials by some largetextbook companies which carry the title "Enrichment Mathematics"are far too often no more than disguised drill in the form of a puzzle. Ithas been this teacher’s experience that if the materials are truly challeng-


578Mathematics/or Gifted Students

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ing, the average student in a heterogenously grouped class will not enjoyparticipating. Therefore if the average students regularly ask to do the"enrichment," a clue has been provided that the material may not bechallenging enough for the able students. Obviously there can be variouslevels of enrichment for students who have finished their basic assign-ment. Some of this enrichment should be planned activities which willchallenge the best students. Traditionally this has not happened with anygreat regularity.

Using Mathematics to Challenge

Mathematics provides an excellent vehicle for providing challenging ac-tivities for GCT elementary and middle school students. Mathematics ac-tivities can be classified into two broad categories, i.e., 1) short challeng-ing experiences, and 2) indepth experiences which teach specific mathe-matics objectives. The second category provides the challenging activitiesof number one, but it goes much deeper. It needs to be clearly stated thatthere is nothing wrong with activities that have the main objective of pro-viding a challenge to the student; however, activities that only challengeare not adequate in and by themselves. There seems to be considerabletemptation for those working with GCT students to settle only for the ac-


Mathematics for Gifted Students 579

tivities which have only the objectives of being challenging. The teacherof GCT students need numerous short activities to use when one studentfinishes before the others and to fill in other short time gaps in the classperiod. Any teacher of GCT students is fully aware of the extreme abilityrange within the group, and knows that equal progress is impossible.

Category I: Short, Challenging Experiences

The following Category One examples are presented to describe to thereader what is meant by short, challenging experiences. These activitiesform a group of problems that are fun�don’t be surprised if you end updoing them all before moving to the next section.

On examples a through c, look for a pattern before replacing the "?" witha numeral. The answers are at the end of the article.

vw.

c. 1,3,8, 19,42,

d. 24, 17,50, 1,2,5,26,37,

e. Each letter represents a numeral. What are these numerals?

CCB+ CCAABDA

f. A gerbil, large cat, a goat, and a cow are named Sandy, Sam, Jim, andBetty. Read the clues below to match names and animals.

1. Jim is smaller than either the cat or Betty.2. The cow is younger than Sandy.3. Sam is the oldest and is a friend of the cat.

Hint: Using the strategy of making a diagram will aid in solving the prob-lem. Place a "0" in spaces which are "no" and a "+ " in spaceswhich are "yes."


580Mathematics/or Gifted Students

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g. Remove 6 toothpicks and leave 2 squares. (The trick is that the two squareswill not be the same size.)

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13 14 15

10 16 17

Teachers who use these and similar activities would correctly state thatthey teach logical thinking, which no one would oppose. It has been thisteacher’s experience that far too many activities of this type are used withthe quick thinking mathematics student. The fact that the highly capablestudents can do them more quickly than the average students, coupledwith the fact that the former group likes them and asks for more, seem toconvince teachers of GCT students that the enriched curriculum shouldbe heavily weighted with this type problem. Problems of this type alsoadapt well to the ditto and hand out procedure.

Category II: Indepth Mathematics

Teachers of gifted, creative, and talented students who rely too heavilyon Category I activities and who are not taking advantage of far reachingor expansive activities in mathematics are passing up great opportunitiesto provide exciting experiences and an indepth encounter with mathemat-ics for these exceptional students.The general topic of geometry abounds with indepth mathematics

which is far too often neglected. Even though we think that area, perime-



ter, and volume are basic concepts, teachers should determine by pretestsif the able students possess the understandings and skills to comfortablysolve measurement problems. The concepts of area and perimeter can bedeveloped most effectively by using geoboards. The concept of volumecan be developed by using solid models. The fourth, fifth, and sixthgrade students who need enrichment are soon able to graduate to the for-mulas of metric geometry.Due to space limitations, only a small portion of geometry can be pre-

sented in this article. The concept of Pick’s Theorem presents clearly theidea of indepth mathematics. Even though this lesson is best presentedwith a geoboard, it can be presented with graph paper or evenly spaceddots on a sheet of paper. The directions are given to stretch a rubberband around the designated number of nails with no nail left inside. Thearea of the figure is to then be determined. The drawing shows fourexamples. Table 1 gives four examples with answers completed.

TABLE iNo Nail Within Boundary

Number of Nails FormingBoundary

Area Within Boundary

1P/22

The next step is to ask the student the area enclosed within 7 nails with-out using the geoboard. Students readily discover the pattern and willstate the area to be 2/2. In an attempt to force a student to generalize,ask how much area would be within the boundaries of 23 nails. This can


582Mathematics for Gifted Students

not be done on the geoboard. The student must use one of the higher lev-els of Bloom’s Taxonomy of Learning called synthesis. The generaliza-

tion is Area =number of nails

The directions for the next problem follow: Stretch the rubber bandaround the designated number of nails with one nail inside the boundary.The area of the figure is then to be determined. The accompanying geo-board shows the area within the boundary of 5 nails. Table 2 containsthree examples with answers.

TABLE 2One Nail Within Boundary

Number of Nails FormingBoundary

567

Area Within Boundary

2’/23

3’/2

What is the generalization?Other problems can be used asking for area with 2 nails, 3 nails, etc.

within the boundary. Being able to form generalizations is a powerfultool, one I believe is not frequently taught. Teachers of GCT studentsmust develop teaching objectives which seek the outcome of forminggeneralizations.Another procedure for teaching the ability to generalize utilizes the

Tower of Hanoi. This aid and the preceding geoboard can readily bemade from wood. The objective of the activity is to move the discs one ata time from one peg to another in the least number of moves without everputting a larger disc on a smaller disc.



Table 3 shows the number of moves needed for 2, 3, and 4 discs.

TABLE 3Moves Required for Tower Game

Number of Discs Number of Moves

Students will readily see the pattern developing of doubling the num-ber of moves last made and adding one. The "heavy stuff" comes whenthe student is able to formulate a generalization of how many moves itwould take for 15 discs.Another Category II activity that has been very popular with talented

students is called "Sum Buildings." The accompanying playing board isneeded. The directions are: Each player selects a different sum, e.g. 10,12, or 8, that will be the sum of the building. The leader rolls two dice todetermine the sum and who will get to place a marker on one of theirnumbers. The first person to reach the top wins.

For the less curious students, this i^nly a game. For the able studentswho are capable of analyzing a situation, it is a game of chance. If en-couraged and given the opportunity, the game will draw them deeply intothe various combinations that dice produce. The final conclusion will bethat out of 36 throws, one can expect the dice to come up seven more of-ten than any other. By chance alone, the persons choosing the sevenbuilding and two building will have equal chances of winning. Studentsin this teacher’s class discovered that with cheap dice the small numberedsides are heavier and since the six is opposite the one, the six comes upmore often than the one.The following diagram shows the mathematical combinations of unbi-

ased dice.


584 Mathematics/or Gifted Students

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A parent of a student who was accelerated in junior and senior highschool mathematics recently told this teacher that his daughter, who isnow in college, is experiencing difficulty in working with percent. On thesurface this may seem unbelieveable. However, the last class instructionthe young woman would have had in consumer mathematics would havebeen in the sixth and seventh grades. It is extremely important for teach-ers of fifth, sixth, and seventh ^de GCT students to make sure theyhave ample opportunity to work with the more challenging types of per-cent problems and other mathematics involving life skills.

Example: The Hub Clothing Store has a 25% sale on all items and is allow-ing 5% of the sale price for cash. What would the price be to a cus-tomer who pays cash for a dress regularly priced at $35?

Many times students who are able to handle percents between one and99 have difficulty computing with percents less than one and more than100. Teachers of talented students should make sure these concepts are


Mathematics for Gifted Students 585

understood. When students start the algebra/geometry sequence, thedays of formal training in consumer mathematics are over. Even as capa-ble as these students are in mathematics, it cannot be left to chance forthem to learn life skills in mathematics on their own.

Example: Due to the increases caused by inflation, a family will have tobudget 117 percent of the $13,750 they spent last year. What mustthey budget?

Example: The baby elephant was increasing in weight at the rate of Vi per-cent each week. What would a 500 kilogram baby elephant weighafter one week of growth?

Summary

Even though short, challenging activities as characterized in this articleas Category I are desired, they should never be considered sufficient inthemselves for the total program. Their advantages are that they areshort, challenging, and can nicely be used to fill in short time periods.

<(. . . activities that only challenge are not adequate in andby themselves.? 9

________

Mathematics for GCT students can not be reduced to a handout withdirections to be placed in the hands of students. They need planned de-velopmental activities, just as do other students. The teacher of these stu-dents must have knowledge and a supply of the indepth mathematics asdescribed as Category II, which can be used to draw the student deeperand deeper into mathematics. The examples presented in this article areonly a sample of the many that are available.No one ever said that meeting the needs of fast learners would be easy.

However, if the GCT movement is ever to achieve the tremendous poten-tial that is possible, teachers of these students must 1) have the mathe-matical knowledge needed to challenge the students and 2) have themathematical activities that will pull students deeply into mathematics.

Answers to Exercises

a. 9 b. 2 c. 89 d. 50


586Mathematicsfor Gifted Students

e. A =1f.B =0’

D =6"(5tGootMCow

g. Remove toothpicks numbered 11, 13, 14, 16,4,5.

^ , ,^ . . number of nailsGeoboard Exercises: Area = ��

2

Tower of Hanoi Exercise: Moves = 215 -1Clothing Exercises: $21Budget Exercise: $16,087.50Elephant Exercise: 502.5 kilograms

REFERENCES

1. MARLAND, S. P. Education of the Gifted and Talented: Report to the Congress of theUnited States. Washington D.C.: U.S. Government Printing Office, 1972, pp. 25 &26.

2. UNDERHILL, BOB. Teaching Elementary School Mathematics. Columbus, Ohio: CharlesE. Men-ill, 1981, p. 17.

Ray KurtzKansas State UniversityCurriculum and InstructionManhattan, Kansas 66506

CARBON DUST

Carbon dust from beyond the solar system has been found in a 4.5 billion-year-old meteorite, by scientists from The University of Chicago and CambridgeUniversity.

R. S. Lewis and E. Anders of Chicago and P. K. Swart, M. M. Grady, andC. T. Pillinger of Cambridge reported their findings in the April 1983 issue ofScience magazine.

Interstellar dust grains are thought to provide the raw material for the forma-tion of new stars and planets. They are common in the space between stars andhave for decades been studied by astronomers with optical and radio telescopes.The interstellar dust grains could be identified because the contained twice as

much "heavy" carbon (carbon with atomic weight 13) as is found on Earth or inthe rest of the solar system.


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Mathematics for Elementary and Middle School Gifted Students