Teaching Math - Multiple Intelligence Students
Since Howard Gardner's initial articulation of his Multiple Intelligence Theory (MIT), academic reform proponents have embraced the notion that students are intelligent to varying degrees in different, interconnected areas. Gardner has asserted that the framing of intelligence as specific to the individual is a relatively new concept in Western society, and student success is rooted in his or her own, unique interpretation of the material. By extension, teachers are facilitators of learning insofar as they utilize strategies that correspond to multiple intelligences.The Activities: Mathematics for Seven Intelligences
While Gardner identified eight, human intelligences, this particular inquiry asserts general, mathematics activities for seven intelligence areas (McClellan and Conti 213; Vardin 40); namely, logical-mathematical, kinesthetic, verbal-linguistic, interpersonal, intrapersonal, naturalist, and visual-spatial. Specific activities for the seven, intelligence areas will be recommended for each of the eight, following, content areas: Exploring real numbers, adding real numbers, subtracting real numbers, multiplying real numbers, dividing real numbers, properties of real numbers, order of operations, and variables. Though the activities are designed with the aim of enhancing achievement for ninth grade students who scored between the 25th and 45th percentile on the Illinois Standards Achievement Test (ISAT), they are concurrently founded on the empirical evidence demonstrating that integrating the intelligences within a classroom yields more positive results than segregating classrooms according to intelligence style (Hill 127).
Logical-Mathematical IntelligenceOften more simplistically labeled as scientific thinking, logical-mathematical intelligence supports, above all else according to MIT, efficient problem solving (Gardner, 19-20). An individual that embodies this type of intelligence is able to organize and prioritize multiple variables and formulate hypotheses with the specific aim of solving a problem; this realm is indicative of nonverbal intelligence. Gardner writes in his text entitled Multiple Intelligences: The Theory in Practice that "a solution to a problem can be constructed before it is articulated. In fact, the solution process may be totally invisible, even to the problem solver" (20). The crux of IQ tests and comprehensive, standardized assessments are devoted to evaluating logical-mathematical reason in conjunction with language skills. Undoubtedly, an individual who holds this form of intelligence is predisposed toward mathematical achievement. However, when this individual is failing to reflect his or her logical-mathematical knowledge in the classroom or on tests such as the ISAT, there are activities that can facilitate the use of his or her efficient, problem-solving abilities.
Exploring Real NumbersDuring an introduction to real numbers, the goal for students with logical-mathematical intelligence would be to channel knowledge through minimal verbal activity (Adams 86). For these abstract thinkers, perceiving comparisons between fractions, decimals, and percentages is relatively simple, as they are naturally inclined to sort, order, and compare bodies of data. Specifically, introducing students to real numbers by presenting them with related figures, such as a simple 1/3, .33, and 33%, and asking them to ascertain the relationship between these figures would facilitate learning for logical-mathematical learners.
Adding Real NumbersBy extension, the act of adding real numbers for logical-mathematical students could be presented as a means to solving a more complex, numerical problem. For instance and using the above example, students could be asked the following: "If you were trying to add 1/3 and 33%, what would be the optimum ways in which to communicate the problem?" Students should be unfailingly encouraged to make hypotheses and test their ideas, even if they may yield results that are less than ideal.
Subtracting Real NumbersLogical-mathematical students are generally adept at databases, spreadsheets, and other, similar software that channels order. Logical-Mathematical students could make use of Microsoft Excel in order to explore real number subtraction. In groups, students could be given a problem, then charged to set up formulas within the cells to automatically subtract decimals, fractions, or whole numbers from one another. Groups could then defend their spreadsheet organization as the optimum way in which to find a solution to the problem.
Multiplying Real NumbersTechnology can assist in learning multiplication skills with respect to real numbers as well, as logical-mathematical students can conceptualize how addition relates to multiplication and reflect this on a calculator or software program (Wills and Johnson 260). For instance, the repeated action of adding 1/3 + 1/3 + 1/3 to yield 1 can be presented as tantamount to 3 X 1/3= 1. Of course, more complex multiplication problems, potentially using negative numbers or lengthy decimals, can be taught this way as well.
Dividing Real NumbersThe connection between multiplication and division is one that can be presented to logical-mathematical learners with relative ease. For instance, learners may be able to make the following inferences: "If 10 X 9 = 90, then 9 X 9 = 81 because 90 - 9 = 81; If 90 / 9 = 10, then 81 / 9 = 9 because 90 - 81 is 9;" such methods support a problem-solving approach to division that is more effective than memorization, particularly for logical-mathematical learners (Wills and Johnson 260).
Properties of Real NumbersIn order to present logical-mathematical students with the commutative, associative, distributive, density, and identity properties of real numbers, the teacher could present the natural problem-solvers with examples of the properties (ie. 3 (2+3) = 15 for distributive) and charge them to ascertain how the properties work on their own. Such activities would organically lead into activities related to the order of operations.
Order of OperationsUnlike linguistic learners who benefit consistently from verbalizing mathematical concepts, logical-mathematical learners conversely benefit from maintaining the purity of numbers. In teaching the order of operations, teachers could present students, or groups of students, with the elements of the order (ie. groups, exponents, multiplication, division, addition, and subtraction) in no particular order, problems that encompass all of those elements, and the correct answer, charging students to figure out the order of operations on their own and defend their position.
VariablesVariables are most frequently introduced through word problems, as teachers ask students to identify the unknown element and represent it as X. However, for logical-mathematical learners, the identification of the unknown is one of the most salient tasks, as the latter operations that allow the student to solve for the variable may come easily. Teachers could present students with a visible scenario, as opposed to a word problem on a worksheet, such as giving Student A five dollars, giving Student B an undisclosed amount, and giving student C twenty dollars. Students could then formulate hypotheses regarding how to possibly set up and solve a related equation.
Kinesthetic IntelligenceMIT posits that kinesthetic intelligence is related to the cognitive strength of movement (Gardner, 23); this is a form of problem solving in that the body ascertains how to best solve a problem, such as expressing an emotion through dance or hitting a baseball, with movement. In either case, calculations are made and movement is enacted accordingly. The amount of kinesthetic intelligence, undoubtedly, varies considerably between individuals, and this form of intelligence presents the greatest problem within small, overcrowded classrooms that inherently oppose movement.
Exploring Real NumbersA core method for facilitating learning for bodily-kinesthetic learners is to integrate physical movement and tactile sensation through as many channels as possible (Adams 86). When exploring real numbers, students should be encouraged to act out example problems through dramatization (Adams 86). For instance, the teacher could use a line of students to represent a number line, giving a certain color hat to "negative" students, and having the class explore how spaces on a number line are represented (ie. 3-5 = -2 on a number line).
Adding Real NumbersSimilar simulation could be utilized during the addition of real numbers, as could the association of certain movements with the act of addition. For instance, students could raise both hands in the air while learning about key concepts in the addition of real numbers; this could be mimicked on an exam to foster memory.
Subtracting Real NumbersSubtraction concepts could then be conversely represented by dropping hands below the waist or another movement that could be used to trigger memory. For instance, in asserting that -2 - 3 = -5, the teacher could charge students to drop hands down a level for each negative sign to indicate that the number is becoming increasingly negative.
Multiplying Real NumbersTeaching both multiplication and division using groups of people is a salient way in which to facilitate these operations for bodily-kinesthetic learners. Grouping students, for instance, into three groups of two and then charging them to articulate the solution to 3 X 2 is a simple example, but can be utilized within a myriad of contexts.
Dividing Real NumbersUsing the same example, having the groups reflect on 6 / 2 using their dramatization is a way in which to engage students in physically embodying mathematical operations. A group of six, for instance, can be factored into two groups of three or three groups of two.
Properties of Real NumbersKeeping the students moving and in groups, properties of real numbers can be taught by adding objects into the dramatization. For instance, in demonstrating the distributive property using the equation 3 (4 X 3 X 2) = 72, three groups of students could be passed a "transforming ball" that represents the number 3 in succession, with the "4" group being changed to a "12" group, the "3" group being changed to a "9" group, and finally the "2" group becoming a "6" group after being passed the ball.
Order of OperationsOrders of operations could be taught by having a group of five students represent various parts of an equation. For instance, an equation of 3(X-4) + 12 = 15 would have one student represent "3", another represent "X," and so on. The seated students can discuss which operation should be performed first in order to solve the equation.
VariablesIntroducing variables for bodily-kinesthetic students can be done using objects, similar to the activity for teaching properties of real numbers. A "variable ball" or "ball x" could be used to represent the variable with students representing various parts of an equation. The "equals" sign could be represented with a line, and students could dramatize solving for "ball x." The goal would be to keep students engaged through physical movement.
Verbal-Linguistic IntelligenceLinguistic intelligence is frequently considered on most standardized tests, including IQ tests. However, this type of intelligence often eludes consideration for mathematics curriculum developers. Gardner cites that "the gift of language is universal, and its development in children is strikingly constant across cultures. Even in deaf populations where a manual sign language is not explicitly taught, children will often "invent" their own manual language and use it surreptitiously!" (Gardner 26). For individuals with a particular affinity for language, however, the need to learn through verbal and written communication is paramount.
Exploring Real NumbersA key component of mathematical learning for verbal-linguistic students is word problems. While logical-mathematical students are able to efficiently find solutions to word problems, verbal-linguistic students learn by creating and verbalizing the problem (Adams 86). When exploring real numbers, by extension, it is key to not just present these students with problems but charge them to create their own; these problems need not be limited to a brief paragraph but can be "problem stories" that embody decimals, fractions, integers, and other real number concepts.
Adding Real NumbersFor all operations of real numbers, verbal-linguistic students benefit from keeping a journal that allows them to personalize the learning experience. When introducing the addition of real numbers, for example, the teacher could write a prompt-problem on the board such as "3/4 - (-1/2) = X" and then ask students to first try to solve the problem and then articulate in their journals any problems they encountered and why they believe they encountered them.
Subtracting Real NumbersSimilarly, when subtracting real numbers verbal-linguistic students benefit from verbalizing the learning experience. Outside of the context of their journals, these students can create oral presentations of concepts for their classmates. The teacher could group students into fours and ask them to present a scenario in which a key subtraction concept is reflected.
Multiplying Real NumbersVerbal-linguistic students have an affinity for organizing information into verbal patterns (Adams 86). By extension, when learning multiplication of real numbers, students could be charged to create a presentation for their fellow students that teaches them the key concepts learned; this could be a short or long-term project that could include the creation of a mock "textbook" or worksheets.
Dividing Real NumbersSimilarly, students could be asked to create a "division story" that will be presented to the class. Individually or in small groups, the teacher would stipulate the story had to relate to certain concepts, such as dividing fractions, and then students would have a class period to create their story. Alternatively, students could pull words out of a hat such as "mailman" or "thunderstorm," and the story would need to include these words in addition to the mathematical themes.
Properties of Real NumbersWhen introducing students to real numbers' properties, the students could be grouped and assigned a particular property. They could then be asked to create a study sheet for their assigned property that is no more than one-page and clearly articulates the key concepts of the property; this sheet would then be copied and distributed to all students so that each student would have a sheet on each property. Groups could then recommend two to five problems related to their property to be included on the assessment and the teacher could synthesize a student-made test.
Order of OperationsThough GEMDAS or PEMDAS is the device utilized most frequently for the memorization of the order of operations, verbal-linguistic students could be charged to create another way in which to memorize the order; this could be a song, poem, story, or other manifestation of language. Ideas could be presented to class for a vote on the optimum alternative to GEMDAS.
VariablesVariables are an abstract concept with which verbal-linguistic students can struggle if they are disallowed to process it in their own way. The teacher could introduce variables by asking students to articulate the definition of an unknown in their journals. After responding to the prompt, students could discuss how unknowns surface in real life situations and how algebraic equation could be used for solving those unknowns.
Individuals with interpersonal intelligence are sensitive to distinctions among other people, particularly with respect to emotions, honesty, and intent (Gardner, 23). Teachers, politicians, and counselors usually embody this form of intelligence and the most apparently sociable students may be interpersonally intelligent. Group cohesion, solidarity, and socialization are critical vehicles for learning for students who are interpersonally intelligent.
Exploring Real NumbersInterpersonal students are generally the born leaders in the classroom, and they thrive within group contexts. When exploring real numbers, the teacher could charge students to design a word problem specifically for another student in the classroom using what they know about them (Adams 86). Each student would then have a personalized word problem that may correspond to their likes or dislikes.
Adding Real NumbersCooperative learning that is highly organized is a salient context for interpersonal students. Placing students in small groups and affording each student a specific role, such as the recorder or speaker, and then tasking them with creating addition problems to present to the class would relate directly to those with interpersonal intelligence.
Subtracting Real NumbersBecause interpersonal students excel at interpreting the positions of others, asking them to create a lesson plan or presentation for a specific group of people aids them in learning as well. For instance, when being introduced to subtracting real numbers, students could be charged to, in groups, design a learning exercise for students who struggle with fractions or decimals. They would then be asked to defend their choices.
Multiplying Real NumbersBridging the gap between verbal-linguistic and interpersonal intelligence, an exercise for both types of these students to learn multiplication would be to create a mock-debate over various multiplication concepts (Adams 86). One side of the debate would knowingly be wrong, and the "winning" side would have to prove why their position is correct. Each team would then switch so as not to damage the self-efficacy of students.
Dividing Real NumbersA key concept for these students is collaboration; thus, if the constructs of scheduling allow, students could team with the science class in discussing and defending real-world applications of real numbers' division. For instance, students could brainstorm as many science-related topics as they could think of that utilize division consistently. These ideas would then be presented to the class or to the teacher for assessment.
Properties of Real NumbersIn extending the verbal-linguistic exercise in which students were placed in groups and assigned a property in order to create a study sheet and student-designed test, an interpersonal extension of this exercise would then be to have students reinterpret what they have learned in their journals. For instance, students would reframe what their fellow students were teaching them in their own way, and aid their own group members in understanding that which was taught by other groups.
Order of OperationsA unique exercise for exploring the order of operations is to group students in large groups of six, assigning each student an operation (groups, exponents, etc.). The students then have to defend their operations place in the order. Alternatively, students could be grouped in small teams and assigned an operation (Team Exponent, Team Addition) and then defend their position in the order as a collective to the class.
VariablesUsing the natural leadership skills of interpersonal students, an activity for exploring variables could involve the creation of a presentation for an elementary school student that explains what a variable is and how it is useful. The presentation could involve visuals or dramatization and would aim to simplify the abstract concept of a variable into terms a child could understand.
Intrapersonal IntelligenceIn contrast to interpersonal intelligence, intrapersonal intelligence is, in essence, self-knowledge (Gardner, 25). These students are generally private and may struggle with social situations. In group contexts, by extension, interpersonal students may be submerged under their comparatively more interpersonally intelligent peers.
Exploring Real NumbersIntrapersonally intelligent students thrive in setting their own personal goals for growth, as they are generally cognizant of their abilities and scope of current understanding (Adams 86). When exploring real numbers, students could use their journals to reflect on what they know about certain real number concepts already, and what they hope to learn in the future. The teacher could use these journals in alternative assessments.
Adding Real NumbersWhen learning the addition of real numbers, these students would benefit from exploring the concept on their own initially, which overlaps slightly with how concepts should be introduced to logical-mathematical students (Adams 86). The teacher would present students with a set of problems and ask them to try to find solutions and, if they struggle, ask them to articulate why they are struggling.
Subtracting Real NumbersThese students also thrive on self-reflection. When learning real numbers' subtraction concepts, students could be asked to create a word problem, or story problem, that relates to a personal experience. These should not presented to the class but to the teacher in journal form.
Multiplying Real NumbersPrior to the introduction of multiplication concepts, students should be asked to ascertain ideas about multiplication from what they have already learned. For instance, teachers could prompt students to write in their journals how the following two problems might relate to one another: "1/3 + 1/3+ 1/3" and 3 X 1/3."
Dividing Real NumbersSimilarly, students should use what they have learned about multiplication during any introduction to division, even though the core concepts may diverge from one another. Students could be charged to articulate a real-world scenario that they might encounter that would involve real numbers' division in their journals.
Properties of Real NumbersIntrapersonal students would benefit from creating a "Properties Journal" or "Properties Packet" that serves as a personal study guide. Students would be urged to make the work visually appealing and correspond to their own tastes specifically. Each property could have a page that lists core concepts of the property, a personally applicable word problem, and other relevant information.
Order of OperationsSimilar to the verbal-linguistic exercise for creating an alternative to GEMDAS, intrapersonal students could be asked to create their own device for remembering the order of operations. For instance, a student who likes music might assign band names to each letter or a student who likes sports might assign team names to each letter.
VariablesAn introduction to variables might begin with a self-reflection on what an unknown is. For instance, the teacher could charge students to write in their journals regarding a time when something was unknown in their lives, and how that affected other areas of their life.
Naturalist IntelligenceNaturalist intelligence corresponds to knowledge of the living world. Those with naturalist intelligence are the biologists and environmentalists of the world. These students learn by interacting with nature and can often feel stifled in the classroom space.
Exploring Real NumbersDuring real number exploration, students could be asked to bring in ten to twenty natural objects, such as rocks or shells, and use those to explore fractions, decimals, and other critical aspects of real numbers. Alternatively, the teacher could bring students outside and have them gather their objects on school grounds.
Adding Real NumbersDuring addition, students could write a story about how a biologist, animal, or explorer would have utilized addition in the past. Students could pull words such as "ocean" or "woodcutter" out of a hat for creative inspiration.
Subtracting Real NumbersSimilarly, students could create word problems or story problems that correspond to the natural world for homework and exchange them during class the next day. Students would then be finding solutions to each others' created problems; this could be extended by having the second student who is solving the problems write a related problem designed for the original writer.
Multiplying Real NumbersNaturalistic learners are inclined to investigate and solve problems similar to the way in which logical-mathematical learners are inclined to do the same (Gardner, 48). By extension, having students discover multiplication concepts is conducive to this type of intelligence. Teachers could present students with multiplication problems, such as .33 X .66, and charge them to find the answer first on a calculator and then work backward in order to extricate the steps for discovering the answer.
Dividing Real NumbersWith respect to division, teachers could present naturalistic students with nature-specific word problems or story problems. For instance students could relate cell division to real number division and teach this relationship to the biology class.
Properties of Real NumbersProperties of real numbers can be taught using science-related scenarios as well. For instance, students could be asked to team with the science class or science teacher and discuss real-world instances of when scientists make use of the properties such as within chemistry formulas.
Order of OperationsThe order of operations could be conveyed by unifying naturalistic and visual-spatial intelligence. Students could be asked to construct a representation of GEMDAS using natural objects (ie. twigs, pebbles, etc.). The class could then have an "Order of Operations Nature Exhibition."
VariablesAgain relating back to scientific scenarios, naturalistic learners could discuss when there is an unknown in a scientific experiment and how the scientific method is similar to an equation. Students could then, in groups, endeavor to solve a problem using variables and the scientific method.
Visual-Spatial IntelligenceVisual-spatial intelligence relates to the creative, right brain. These learners are artists, engineers, and designers. They learn by seeing and, by extension, can struggle with abstract, mathematical concepts.
Exploring Real NumbersThe key to catering visual-spatial students is to facilitate creative, visual representation of mathematical concepts in space. While this is relatively simple in the realm of geometry, algebra can present more significant challenges when strategizing for this unique population. When exploring real numbers, charge students to represent concepts in their journals using vivid colors, perhaps red for negative numbers and black for positive numbers; green for proper fractions and orange for improper fractions.
Adding Real NumbersWhen teaching these students about real numbers' addition, task them with creating a poster that would represent key concepts, such as a negative plus a negative equals a negative. Alternatively, groups of students could use physical objects such as blocks or popsicle sticks in order to explore addition concepts (Adams 86).
Subtracting Real NumbersSimilarly, students could prepare a work of visual art that relates to real number subtraction; this could overlap with verbal-linguistic activities in that the piece could have a written, background story and students should be able to defend elements of their piece. For instance, the right side of the sculpture is red because the solution to the problem was negative.
Multiplying Real NumbersWorking backward from the addition activity, students could be introduced to multiplication by exploring concepts visually in their journals and then assigning colors or shapes to key concepts. Students would then search for an artwork that embodies some or all of those elements. For instance, a Picasso painting might represent rules of fractional multiplication for some students.
Dividing Real NumbersMore simply, division could be explored through the use of paper cutting. Students could individually explore how a whole piece of paper might represent 4/4, and four pieces would need to be yielded if that paper was to be divided by four (Adams 86).
Properties of Real NumbersWith respect to properties of real numbers, visual-spatial students might struggle with the abstract concepts unless they can visualize the properties. Students could be placed in small groups or pairs and assigned a property, tasked with visually representing that property and then presenting their finished work to the class.
Order of OperationsStudents could also visually demonstrate an alternative to GEMDAS. In groups, students could select a color or design element that corresponds to each function (ie. bold, red lines for subtraction) and then create a poster that embodies their ideas. Students should then defend their choices to the class or in a written document for assessment.
VariablesA creative introduction to variables could involve a strictly artistic task. Students could individually consider what an unknown is to them; this could be an extension of the intrapersonal, journaling exercise. Following a journaling experience, students would be tasked with creating a work of art that represents an unknown, and titling it accordingly.
Synthesis and SummationCatering to the plurality of intelligences is one of the greatest burdens of twenty-first century mathematics teachers. However, many of the activities described herein correspond or can be extended to two or three intelligence areas. Improving the academic performance of students struggling in mathematics is rooted in strategies that relate to MIT. More saliently, however, constant observation of students and solicitation of feedback ensures that students are genuinely responding to teaching strategies (Lamarche and Bisson 268).
Works CitedAdams, Thomasenia Lott. "Helping Children Learn Mathematics through Multiple Intelligences and Standards for School Mathematics." Childhood Education 77.2: 86.
Gardner, Howard. Intelligence Reframed Multiple Intelligences for the 21st Century. New York: Basic Books.
Gardner, Howard. Multiple Intelligences: The Theory in Practice. New York: Basic Books.
Hill, Denise. "The Mathematics Pathway for All Children." Teaching Children Mathematics Oct: 127-131.
Lamarche-Bisson, Diane. "Learning Styles - What Are They? How Can They Help?." World and I: 268.
McClellan, Joyce A., and Gary J. Conti. "Identifying the Multiple Intelligences of Your Students." Journal of Adult Education 37.1: 13-20.
Vardin, Patricia A. "Montessori and Gardner's Theory of Multiple Intelligences." Montessori Life Winter: 40-43.
Wills, Jody Kenny, and Aostre N. Johnson. "Multiply with MI: Using Multiple Intelligences to Master Multiplication." Teaching Children Mathematics: 260.