Katie sings the multiplication facts quietly toherself. Sam tries to solve problems by tap-ping his pencil in rhythmic patterns. José

draws pictures from his teacher’s tile-array models.Alima sorts her crayons into groups to help herselffind the answer. Lin checks his solutions with hisneighbor. Maria uses logic to extend her knowledgeof simple facts to harder ones. These children natu-rally express their intelligence strengths as theyattempt to master their multiplication facts. Theirteacher knows that children learn in different waysand seeks to consciously translate these differencesinto learning methods that will be meaningful.Howard Gardner’s theory of multiple intelligenceshas significant implications for all mathematicsteachers who are looking for diverse instructional

methods that encourage depth of understand-ing by tapping students’particular inclinations.As Gardner (1991, p. 13) says, “Genuineunderstanding is most likely to emerge and beapparent to others . . . if people possess a num-ber of ways of representing knowledge of aconcept or skill and can move readily back andforth among these forms of knowing.”

In his 1983 book Frames of Mind: TheTheory of Multiple Intelligences, HowardGardner proposed a revolutionary revision ofour thinking about intelligence. Traditionalviews and testing methods emphasize a uni-tary intelligence capacity based on linguistic

and logical-mathematical abilities. Gardner sug-gests that intelligence is based on multiple“frames,” each consisting of unique problem-solving abilities. Gardner’s eight intelligences are

called logical-mathematical, naturalistic, bodily-kinesthetic, linguistic, spatial, interpersonal,intrapersonal, and musical. All people use all theseintelligences in their lives, but through unique rela-tionships between “nature and nurture,” each per-son has a particular mix of intelligence strengths atany given time.

What does multiple-intelligence theory have todo with teaching mathematics? It allows teachersto use eight different possible approaches to math-ematical learning and teaching. This multiple-instruction approach—

• results in a deeper and richer understanding of mathematical concepts through multiple representations;

• enables all students to learn mathematics suc-cessfully and enjoyably;

• allows for a variety of entry points into mathe-matical content;

• focuses on students’ unique strengths, encour-aging a celebration of diversity; and

• supports creative experimentation with mathe-matical ideas.

Because the idea of multiple intelligences is a the-ory, not a strict educational methodology, it can beapplied flexibly and in diverse ways that work forparticular students, teachers, and contexts (seetable 1). Teachers’ strategies may vary from estab-lishing specific times of direct instruction usingvarious methods to setting up multiple-intelligencecenters or stations that students visit at flexibletimes throughout the day. Although all childrenwill benefit from experiences with all intelli-gences, teachers can encourage students to “lean”on their strengths to achieve mathematical under-standing.

The following application of this multiple-intelligence theory to mathematics instructionfocuses on building mastery of multiplication facts

260 TEACHING CHILDREN MATHEMATICS

Jody Kenny Willis andAostre N. Johnson

Jody Kenny Willis, jwillis@smcvt.edu, and Aostre Johnson, ajohnson@smcvt.edu, are col-leagues at Saint Michael’s College in Colchester, VT 05439. Willis’s scholarly interestsinclude mathematics education, pedagogy, and one-room schooling. Johnson’s researchfocuses on curriculum theory and spirituality in education.

Multiply Using Multiple Intelligences

Copyright © 2001 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

as an example. The goal is for children to use theirdifferent intelligence strengths to attain initial con-ceptual understanding of multiplication, then tomove toward developing their own thinking strate-gies for harder facts, gradually building masterythrough practice and problem solving. Multiple-intelligence theory can be adapted to any othermathematical concept or skill.

Logical-MathematicalIntelligenceLogical-mathematical intelligence includes the fivecore areas of (1) classification, (2) comparison, (3)basic numerical operations, (4) inductive anddeductive reasoning, and (5) hypothesis formationand testing—all basic “tools” of the mathemati-cian. Although current classroom practices for

261JANUARY 2001

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262 TEACHING CHILDREN MATHEMATICS

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E 1 Instructional matrix for multiple intelligences in mathematics

Intelligence Materials Learning Activities Teaching Strategies

Logical- calculators creating or solving: worthwhile tasksmathematical manipulatives brain teasers connections with previous

games problems conceptsnumber lines logic puzzles variety of representationsVenn diagrams equations inquiry methods

algorithmsjustifying thinking

Naturalistic natural objects using nature demonstrationsmodels classifying objects outdoor activitiesobservation notebooks observing patterns naturalistic investigationsmagnifying glasses

Bodily manipulatives sequencing movements gestureskinesthetic models exploring tactile models dramatizations

individual children or groups dramatizing hands-on examplesclapping, tapping, hopping physical modelsusing concrete materials

Linguistic children’s books reading word problems storytellingtextbooks writing mathematics stories book cornersaudiotapes listening to explanations humor and jokesactivity sheets talking about strategies questionsjournals assessment tasks

lectureswritten or oral explanations

Spatial computers decorating flash cards mental modelsgraphs drawing diagrams visual cues; e.g., color,charts creating pictures or circles, boxes, arrowsplaying cards other representations guided imagerymanipulatives looking at illustrations graphic organizersdominoes concept maps or websbulletin boardsoverheads

Interpersonal games working cooperatively discussionsshared manipulatives participating in simulations people-based problems

interviewing others peer tutoringengaging in role playing group activitiessharing strategies guest speakersassessing peers’ work

Intrapersonal self-checking materials writing in journals private spacesdiaries or journals addressing values and attitudes choice time

reflecting on connections empowermentwith students’ lives

conducting self-assessment

Musical tape recorders composing, performing, or listening cornersCDs listening to raps, songs, chants rhythmical activitiesinstruments using musical notation background music

creating rhythmic patterns

Source: Adapted from a model by Armstrong (1994, p. 52)

teaching mathematics frequently focus on roteanswers, research encourages children to explorevarious representations for different multiplicativesituations and to investigate the relationshipsamong these models to obtain a deeper mathemat-ical understanding of the operations (Kaput 1989;Kouba and Franklin 1995; Isaacs and Carroll1999).

Logical-mathematical intelligence allows chil-dren to develop an understanding of the three mod-els for multiplication illustrated in figure 1 and toapply this knowledge to real-life situations. Teach-ers usually introduce facts using repeated additionwith manipulatives, such as buttons, candies, orCuisenaire rods. Other possibilities include usingnumber lines or a calculator’s repeated-additionfunction by pressing + 4 + 4 + 4. The array model,which relates to area, can be illustrated with colortiles, geoboards, pegboards, and graph paper. Thecombination, or Cartesian-product, model of mul-tiplication requires each member of a group for onenumber to be paired with each member of a groupfor the second number. For example, four bears ofdifferent colors can be paired with three differentT-shirts in twelve possible combinations in a treediagram.

Effective use of logical-mathematical intelli-gence would enable children to conceptualize mul-

tiplication’s relationship to other operations,namely, as repeated addition and as the inverse ofdivision. This intelligence also underlies the devel-opment and articulation of thinking strategies. Forinstance, students can work from known facts toreason about multiplication problems: “If 10 × 7 =70, then 9 × 7 = 70 – 7 = 63,” or “If 3 × 7 = 21, then6 × 7 would be double that, or 42.” Such strategiesnot only encourage problem-solving skills but alsoare faster than more immature counting methodsand help children learn facts in relation to otherfacts, not as isolated bits of information.

The role of logical-mathematical intelligence inacquiring mathematical skills and concepts is obvi-ous, yet the seven other intelligences also makesignificant contributions as children select theirown unique strategies to move from initial investi-gations, through increasingly sophisticated levelsof understanding, to mastery.

Naturalistic IntelligenceNaturalistic intelligence includes the ability torelate to the natural world with clarity and sensitiv-ity; to recognize and classify living things, naturalobjects, and patterns; and to use the informationgained productively. Ecological and environmentalperspectives are grounded in this type of intelli-

263JANUARY 2001

FIG

UR

E 1 Three models for multiplication showing the fact 3 ×× 4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

(a) Two examples of the repeated-addition model

Striped shirt

Plaid shirt

Blue bear

Red bear

Yellow bear

Green bear

Plain shirt

Blue bear

Red bear

Yellow bear

Green bear

Blue bear

Red bear

Yellow bear

Green bear

(c) Combination model(b) Array model

264 TEACHING CHILDREN MATHEMATICS

gence and use mathematical concepts in their theo-ries and methods. Students who have a particularecological strength show a strong interest in thenatural sciences and in the outdoors.

Children’s first examples of groups of objectsemerge from the “natural” world around them. Forexample, a baby notices the five fingers on each ofher two hands and the two eyes of her father. Initialinformal multiplication experiences spring appro-priately from nature. For example, how many eyesdo the four people in a family have? How manynoses, ears, hands, or fingers do they have? Allmathematics curricula connected with this intelli-gence use as many natural objects and as muchtime in nature as possible. These curricula provideunique opportunities to encourage a sense of won-der at the extraordinary mathematical patternsfound in the natural world—in plants, pets, insects,birds, shells, zoo animals, and the children them-selves. Biological classification systems allow stu-

dents to work in authentic ways on subtle and com-plex organization and numbering. Sea Squares(Hulme 1991) is an example of an effective picturebook for multiplication ideas. The book uses illus-trations of natural creatures to pose multiplicationquestions, such as how many arms are found onfour starfish or how many legs on six pelicans.Teachers and students can create their own prob-lems and examples using number patterns innature.

Bodily-KinestheticIntelligenceChildren with strong bodily-kinesthetic intelli-gence use their bodies in highly differentiated waysto develop and express concepts. They may natu-rally tap out facts with their fingers, trace facts withtheir fingers on other body parts, or manipulatevarious physical objects to represent different facts.Instructional approaches that use a variety ofmanipulatives allow for the expression of bothvisual and bodily-kinesthetic intelligences. Evenadult mathematicians might use this intelligence asthey build models in their scholarly pursuits. Ein-stein described his theorizing as more “visual andmotor” or “muscular” than linguistic (Einstein1952, p. 43).

Teachers who wish to help students acquiremultiplication facts using bodily-kinesthetic intel-ligence may have groups of students dramatizefacts by asking them to act out various problems.For example, teachers could introduce the conceptof multiplication by physically arranging studentsinto three basketball teams with five players each.Students can also do “multiplication exercises” byhopping, jumping, or clapping to various facts. Thefact 3 × 4 might be four jumps—rest—fourjumps—rest—four jumps. “How many jumps inall?” Children can create kinesthetic patterns asthey count to the number 40, stamping on multiplesof 4 and clapping on every number in between.Tracing facts in sand, doing finger painting, orwriting facts in glue and decorating the numbers

FIG

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E 2 Paper-folding task for the facts for 5

FIG

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E 3 Visual models of square facts, rectangular facts, and

prime numbers

4 × 4

3 × 4

1 × 7

265JANUARY 2001

are other possible bodily-kinesthetic activities.Using the picture book Knots on a Counting Rope(Martin and Archambault 1990) as a model, chil-dren can create ropes that represent different factsby knotting strings or by stringing beads.

Linguistic IntelligenceLinguistic intelligence encompasses a wide rangeof language skills, from sensitivity to the meaningsof specific terms to the ability to use language in avariety of contexts. The NCTM recognizes theimportance of language in its Curriculum andEvaluation Standards for School Mathematics(1989), which state that “representing, discussing,reading, writing, and listening to mathematics are avital part of learning and using mathematics” (p.26). Most teachers require children to listen andread during their mathematics lessons; in recentyears, many teachers have begun to initiate moreopportunities for children to speak and write abouttheir experiences.

All children should be exposed to a wide varietyof worthwhile verbal tasks as they encounter theidea of multiplication and learn their facts. Theteacher may offer a persuasive verbal introductionto the exciting uses of multiplication in the world,from the kitchen to the workplace. Children canalso tell or write their own multiplication stories.Many children’s books, such as Anno’s MultiplyingJar (Anno 1983) and One Hundred Hungry Ants(Pinczes 1993), relate to multiplication eitherdirectly or indirectly. Although the text in booksmay use a verbal-linguistic approach to pose prob-lems or offer explanations, the pictures in these

books will also engage students who prefer visualmodels.

Oral recitations of facts may be beneficial forsome students. The facts for 2 and 5 tend to be easyfor children who have been introduced to skipcounting at earlier ages. Children can use languageto communicate the results of multiplication tasks,to explain the reasoning behind their problem solv-ing, to comprehend other children’s thinkingstrategies, and to assess their progress by explain-ing to their teacher, parent, or another student thefacts that they know well and those that they needto practice.

Spatial IntelligenceChildren who have strong spatial intelligence areable to perceive the visual world accurately, createimages in their minds in the absence of physicalstimuli, and produce effective two-and three-dimensional representations. Many adult mathe-maticians rely heavily on this intelligence as theytheorize and represent their findings.

Visual models enhance memory by creatingimages that the mind stores for subsequent refer-ence. Many picture books contain excellentprospects for teachers to develop spatial intelli-gence in students. For instance, in the book Made-line (Bemelmans 1939), the picture of six girls ineach of two straight lines presents an interestingchance to explore whether the girls could begrouped in other ways, such as in three lines offour. Flash cards can be decorated by using differ-ent colors or pictures or by writing difficult facts increative ways, such as in “fat” numerals. Students

FIG

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E 4 The commutative and distributive properties of multiplication

7 × 3 = (5 + 2) × 37 × 3 = (5 × 3) + (2 × 3)

(a) Commutative property (b) Distributive property

3 × 5 = 5 × 3

266 TEACHING CHILDREN MATHEMATICS

could create an advertisement for a multiplicationfact, draw a fact picture, or create a bulletin boardthat illustrates the doubles facts. Children can alsowork with models, such as an egg carton for 2 × 6or a spider’s legs for 2 × 4. A paper-folding taskcould depict the repeated-addition model as eachsegment unfolds. Figure 2 shows such a task forthe facts for the number 5.

Spatial intelligence also fosters deeper under-standing as children solve problems and representsolutions. Children can explore the shapes that areformed as they create “boxes,” or array models,using colored tiles. They will see that many num-bers yield several rectangular shapes, whereasprime numbers yield only one row of tiles (see fig.3). Teachers should guide the process by asking,“Do the multiplication facts that you have modeledalways form rectangles? Why are some facts called

square facts?” The com-mutative property canbe demonstrated byrotating the tile models,and the distributiveproperty can be shownby splitting an array, asshown in figure 4.

Another spatial activ-ity that gives practiceand involves problemsolving is to have chil-dren color multiples ofa number on a hundredchart and look for pat-terns. Children can dothis activity for multi-ples of 2 through 9 and

compare the patterns that are formed. If this activ-ity is done on transparencies, the children can dis-cover common multiples by placing one sheet ontop of another (see fig. 5).

InterpersonalIntelligenceInterpersonal intelligence encompasses under-standing and communicating sensitively with otherpeople. It consists of the ability to notice and makedistinctions about people’s moods, temperaments,motivations, and intentions—and to make relevantdecisions based on this knowledge. Children whoare adept in interpersonal intelligence may thriveon collaborative problem solving, but compellingreasons exist to cultivate this form of intelligencein all students. Nurturing students’ social and emo-tional skills now should produce members of soci-ety who are more caring and cooperative in thefuture. In the world of adult mathematicians, schol-

arship and work are increasingly communal, ratherthan individualistic, enterprises.

Obviously, learning methods designed toenhance interpersonal intelligence focus on coop-eration among students. Even when students workindividually on problems and assignments, theycan be asked to justify their methods and resultswith one another. Partners can work on shared mul-tiplication problem solving or assess each other’swork. Peer tutoring allows students who have moreadvanced mathematics skills to assist those whoseskills are not so advanced, resulting in improve-ment for both. Children can collaborate in short- orlong-term mathematics groups.

Real-world multiplication problems can be con-structed using familiar people and situations. Stu-dents and teachers might invent multiplicationexamples that reflect their interpersonal interests.For example, Alima could make up a probleminvolving the amount of candy collected as she andher best friends trick-or-treat on Halloween. Stu-dents can also interview adults in various profes-sions about how they use multiplication skills onthe job.

IntrapersonalIntelligenceIntrapersonal intelligence is the ability to under-stand oneself—to recognize one’s own feelings, setgoals, and make relevant decisions. In an era thatpromotes collaboration, we must also make surethat we do not neglect the students who prefer towork alone. Intrapersonal intelligence thrives onquiet spaces and personally challenging problems,as well as time for in-depth explorations. All stu-dents benefit from learning to deepen their knowl-edge of themselves, their introspection, and theirability to develop appropriate personal strategies.

This intelligence has significant implications forthe mathematics curriculum. A curriculumdesigned to facilitate a metacognitive perspectivewill emphasize personalized experimentation withdiverse and meaningful problems. The teacher canintroduce multiplication to students as a “big idea”that has extraordinary possibilities for relating toreal and imaginary events in their lives. She canshow how multiplication is used to address issuesof concern to students, such as caring for pets orsaving endangered species. Students should beencouraged to set goals for themselves on manylevels, such as using their choice of intelligences tomemorize multiplication facts, creating their ownpersonally relevant multiplication problems tosolve, and finding or communicating unique solu-tions to teacher-posed problems. Students can alsoassess their own mastery of facts, asking them-

In an era that

promotes

collaboration, make

sure not to neglect

students who prefer

to work alone

267JANUARY 2001

selves, “Which facts do I know well; which onesdo I need to practice?” Finally, students can keepongoing mathematics journals in which theyrecord their multiplication goals, activities, andself-assessments, as well as their attitudes.

Musical IntelligencePitch, rhythm, tonal quality, and emotional expres-sion are all aspects of musical intelligence. Theunderlying rhythmical patterns of music arefounded in mathematics, and the musical notationsystem is based on fractions and numerical pat-terns. Gardner (1983) notes that many composershave been sensitive to mathematical patterns andthat many mathematicians and scientists areattracted to music.

In learning multiplication, students can userhythmic groupings of sound to represent problemsand solutions. These activities can be either kines-thetic, involving hopping or clapping, as men-tioned earlier, or tonal, using the voice or musicalinstruments. Many adults have learned the alpha-bet through singing, and multiplication facts canlikewise be remembered through music. Althoughsome children may be able to compose their owntunes as they sing their multiplication tables,adapting the facts to a familiar tune may be easierfor many children. The familiar song “Row, Row,Row Your Boat” could be adapted to “Two and twoare four. Four and four are eight. Eight and eightare sixteen. Other facts we’ll state.” Students couldcompose and perform similar verses. Some stu-dents may prefer chants or raps. Learning the factsfor 7 are often difficult for children. The rap shownin figure 6 serves as an example of a device thatmight be created to help children remember thefacts for 7.

ConclusionAccording to Gardner’s theory, children are intelli-gent in multiple ways and have unique combina-tions of intelligence strengths. All these intelli-gences can be used throughout the mathematicscurriculum. Rather than treat multiplication asmemorization of facts or rote computation of irrel-evant equations, multiple-intelligence theoryenables students to understand it as an exciting,

FIG

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E 6 The facts for 7 put to song

Let’s rap our facts quickly, Then mathematicians are what we’ll be.For 0 × 7 here’s a tip:0 times a number equals zip.1 × 7 is easy to remember;1 times anything is just the number.Doubles facts are easily seen.Two 7s always equal 14.3 × 7 gives 21.Keep rapping our facts until we’re done.We love math and really can’t waitTo state four 7s is 28.Five 7s gives 35 real quick,But seven 5s is 35, too! Neat trick!Six 7s is double three 7s, 21,So 6 × 7 is 42—such fun!Keep on rapping; we’re doing just fine.7 × 7 equals 49.56 answers 8 × 7.Knowing our facts takes us to heaven.Ten 7s is 70.So nine 7s, one less, equals 63.Mathematicians—that’s what we are.We can rap our facts both near and far!

FIG

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E 5 Patterns of multiples on a hundred chart using the facts for 3 and 4 and a combination of the two fact patterns

0

10

20

31

40

50

61

70

80

91 92 94 95 97 98

82 83 85 86 88 89

71 73 74 76 77 79

62 64 65 67 68

52 53 55 56 58 59

41 43 44 46 47 49

32 34 35 37 38

22 23 25 26 28 29

11 13 14 16 17 19

1 2 4 5 7 8 0

10

31

50 51

61

70

81

91 9390 94 95 97 98 99

82 83 85 86 87 89

71 73 74 75 77 78 79

62 63 65 66 67 69

53 55 5754 58 59

41 43 4542 46 47 49

3433 35 37 38 39

2221

30

23 25 26 27 29

11 13 14 15 17 18 19

1 2 3 5 6 7 9 0

10

31

50

61

70

91 94 95 97 98

82 83 85 86 89

71 73 74 77 79

62 65 67

53 55 58 59

41 43 46 47 49

34 35 3837

22 23 25 26 29

11 13 14 17 19

1 2 5 7

+

Multiples of 3 Multiples of 4 Combination pattern

(Continued on page 269)

relevant way of symbolizing a significant propertyof the world around them. When teachers encour-age the use of diverse intelligence strengths in mul-tiplication, they allow students to increase theircapacities to learn facts by heart, conceptualize themeaning of multiplication, develop thinking strate-gies, solve problems, and engage intensively andcreatively in mathematics. As they do so, studentshave the opportunity to discover and celebrate theirown, and one another’s, unique abilities.

ReferencesAnno, Mitsumasa. Anno’s Mysterious Multiplying Jar. New

York: Philomel Books, 1983.Armstrong, Thomas. Multiple Intelligences in the Classroom.

Alexandria, Va.: Association for Supervision and Curricu-lum Development, 1994.

Bemelmans, Ludwig. Madeline. New York: Viking Press 1939.Einstein, Albert. “Letter to Jacques Hadamard.” In The Creative

Process, edited by Brewster Ghiselin, pp. 43–44. Berkeley,Calif.: University of California Press, 1952.

Gardner, Howard. Frames of Mind: The Theory of MultipleIntelligences. New York: Basic Books, 1983.

———. The Unschooled Mind: How Children Think and HowSchools Should Teach. New York: Basic Books, 1991.

Hulme, Joy N. Sea Squares. New York: Hyperion Books, 1991.

Isaacs, Andrew C., and William M. Carroll. “Strategies forBasic-Facts Instruction.” Teaching Children Mathematics 5(May 1999): 508–15.

Kaput, James J. “Supporting Concrete Visual Thinking in Mul-tiplicative Reasoning: Difficulties and Opportunities.” Focuson Learning Problems in Mathematics 11 (winter 1989):35–47.

Kouba, Vicky, and Kathy Franklin. “Multiplication and Divi-sion: Sense Making and Meaning.” Teaching ChildrenMathematics 1 (May 1995): 574–77.

Martin, Bill, Jr., and John Archambault. Knots on a CountingRope. New York: Trumpet Club, 1990.

National Council of Teachers of Mathematics (NCTM). Cur-riculum and Evaluation Standards for School Mathematics.Reston, Va.: NCTM, 1989.

Pinczes, Elinor. One Hundred Hungry Ants. New York:Houghton Mifflin Co., 1993. ▲

269JANUARY 2001

Do You HaveSomething to Add?Share with readers and the Editorial Panel your opin-ions about any of the articles or departments appear-ing in this issue by writing to “Readers’ Exchange,”NCTM, 1906 Association Drive, Reston, VA 20191-9988; or by sending e-mail to tcm@nctm.org.

Multiply with MI(Continued from page 267)