implications of right/left brain research for mathematics educators

118

Implications of Right/Left BrainResearch for Mathematics EducatorsJohn L. CreswellDept. of Curriculum and InstructionUniversity of HoustonHouston, Texas 77004

Claire GiffordEctor County Public SchoolsOdessa, TX 79760

Debbie HuffmanDept. of Curriculum & InstructionUniversity of HoustonHouston, TX 77004

". . . the left hemisphere primarily controls the rightside of the body while the right hemisphere primarilycontrols the left side of the body. . /)

Over 150 years of clinical evidence have indicated asymmetrical functioningof the brain, but this was largely ignored by all but specialists until theremarkable brain research of the 1960s. The split-brain research, which wascatapulted into the forefront by the award of the Nobel Prize to RogerSperry, captured the imagination of practitioners in every field.The brain is composed of two mirror-image hemispheres and several

commissures, e.g., bands of nerve fiber which connect the hemispheres. Thelargest of these commissures is the corpus callosum. It is now known that theleft hemisphere primarily controls the right side of the body while the righthemisphere primarily controls the left side of the body (the contralateral rule)and the commissures provide communication between the two hemispheres.Additionally, it appears that the two hemispheres respond differentially todifferent kinds of stimuli and that the modes of processing information maydiffer between the two hemispheres.

In normal brains, the corpus callosum, the commissure, the part of thebrain which connects the two hemispheres, "fires’* so rapidly that everythinghappening in one hemisphere is immediately processed in the other. If thisdid not happen, the left hand literally would not know what the right handwas doing. Consequently, even though each hemisphere has certain special-ized functions, the hemispheres share many functions, called bilateral, e.g.,both hemispheres participate in most activities. However, each hemisphereprocesses information differently.One of the hottest topics in the realm of educational (and other) areas is

the concept of cerebral hemispheric asymmetry, e.g., left brain/right brain

School Science and MathematicsVolume 88 (2) February 1988

Left Brain, Right Brain 119

differentiation. Some claim that research findings have broad implications forthe classroom and, indeed, for increasing successful behavior in many liferoles (Hatcher, 1983; Springer and Deutsch, 1981; Johnson, 1982). Otherswarn that actual findings have been far overrated, the current statusunderrated; and they belittle the excitement brain research has spawned(Levy, 1984; Whittrock, 1981). Probably the truth lies somewhere betweenthese two extremes.

What Does Research Say About Left/Right Brain Functions?

Brain research has led to strong medical/scientific support for differentiationbetween the hemispheres in regard to perceptual strengths, skills, andinformation processing approaches (Wheatly, 1977). While this division oflabor is subtle, complex, and not well understood, conclusions of the researchstudies consistently describe the left hemisphere as superior in verbal skillsand the right hemisphere as superior in visuo-spatial skills. Studies that assesshow the brain processes information consistently conclude that the left braintypically functions analytically while the right brain functions holistically.For the purpose of this discussion, cerebral hemispheric asymmetry will be

accepted as a theoretical construct and its implications for mathematicseducation will be explored.The literature repeatedly refers to people as being right-brained or

left-brained. References are made to schools being right-brained or left-brained and even to cultures and subcultures being right-brained orleft-brained (Power, 1984). A "right-brained" child certainly has a functionalleft hemisphere. The distinction is that (1) each will be more sensitive todifferent stimuli than the other; (2) given the same stimulus, one may processthe information in a different manner than the other; and (3) the types ofresponses that are typical of the individuals may differ.Some examples may clarify these three distinctions. Left-brained students

may understand the proof of a geometry theorem after studying definitions,corollaries, and proofs of other related theorems; Right-brained students aremore likely to understand after studying some related geometric pictures(different stimuli). Given a mathematical problem to solve, left-brainedchildren will typically outline the problem very methodically, carefully planeach step and then solve the problem; right-brained children will read theproblem several times, perhaps look at notes and seem to go through an idleperiod while mulling thoughts around, then solve the problem (differentinformation processing). If asked to solve a problem in geometry, left-brainedchildren may well explain how to solve it; right-brained children will morelikely draw a picture of the geometric entity (different response).

Notice that the left-right dichotomy is not a sharp, crisp one. Left-brainedchildren may truly, after reading, have an overview (RB) of the proof;right-brained children will read (LB) and understand the words in the flowchart and geometric pictures. The left-brained children demonstrate spatialperception (RB) by knowing how to solve the problem in geometry;


120 Left Brain, Right Brain

right-brained students may use symbols (LB) on their pictures of thegeometric entity. Both types of children use both hemispheres; the differenceis in preference and dominance.Some individuals have very little apparent hemispheric dominance; others

have a strong left or right-brain tendency. But even with cases of strongdominance, it is not an either/or situation. Some describe it as a continuumwith rarely more than a 60%/40% spread in normal persons (Kail, 1981).Actually, it is more appropriate to consider that the brain has twointerdependent sources of capacity, each of which can be developed.

Virginia Johnson (1982) presents a nicely organized and fairly typical listingas she differentiates the way the two hemispheres perceive information andthe ways they process that information. See the following:

Left Hemisphere

abstract symbolslanguage

readingspeakingwriting

mathematicsnumeralbasic factscomputation

logic2

time�linear

music�rhythm4

Left Hemisphere

logical2sequentialconvergentexclusiveanalyzingeither/or

decision-making

Information PerceptionRight Hemisphere

visual imageslanguage

expression, tone

body language, gesturesfacial expressions

mathematicsshapespatternsrelationships

time�linearcreativity3time�cyclicalmusic�melody4functions5

Information Processing

Right Hemisphere

intuitive

holistic

divergentexclusivesynthesizingmultiple implications

�^ome do not agree that logic is a left-brain function; Levy, for example, contends that geometric

reasoning is logical and right-brained.

creativity is a controversial characteristic. It is repeatedly linked with right-brain activity, but some writers

(maybe left-brainers!) seem very indignant with the limitation. (Ferguson, 1973)^usic is often described as right-brained oriented, but several writers have taken Johnson’s approach

dividing it into rhythm and melody.^his appears to be in conflict with Levy’s research that the LH matched on the basis of function, the RHon the basis of appearance.

In summary, Gaylean (1981) writes that since the left brain must translatethe images of the right brain into words, the left brain functions as the



"alphabet of the mind." Hence, "without the right brain there would be noidea; without the left brain, the idea would not be encoded, understood orcommunicated." A familiar example of this is the dream that cannot beremembered. These statements may be slightly oversimplified.

Are Schools Left-Brained?

Many writers claim that schools stress left-brain oriented activities (Levy,1983; Hatcher, 1983; Johnson, 1982; Kail, 1981). Traditionally, much of theinformation taught has been transmitted by reading, listening, and computa-tional drills. Students typically provide responses in writing. Organizationalskills are usually approached in a sequential fashion�listing, outlining,keeping notebooks, and being punctual. An apparently idle student who,while his classmates are busy doing arithmetic computation, says he is "justthinking" is typically told to get busy.

All too often, there is a minimum of experimental, sensory stimuli; evenscience may be learned only by reading, especially in early grades. Numberconcepts are frequently taught as abstract manipulations of symbols ratherthan by using concrete manipulatives, a right-brained activity. In a study byScott (1983), it was found that few K-5th grade teachers used anymanipulative materials more than five times a year. Indeed, only 60% of firstgrade teachers used manipulatives; the percentage of teachers using manipula-tives steadily declined each year after the first grade. Likewise, the use ofmanipulatives is particularly low when one considers that Weibe (1981) foundthat "actual use of manipulative materials is often less than expected use" (p.62).

Left-brained children, as a group, have not been found to be moreintelligent than right-brained children (Levy, 1983). They do have a tendencyto perform better in school, but that is thought to be because the schoolsetting is more conducive to left-brained behavior and most tests areleft-brained instruments (Hatcher, 1983). Few schools or teachers contrivesettings that allow right-brained strengths to be recognized and fullyappreciated. (This is not surprising since a large majority of teachers andschool administrators are left-brain dominant themselves.) Then, when certain(right-brained) children do poorly, schools and teachers describe the problemas the child’s problem, never considering the interactive aspect of the settingand the personal dimension or recognizing the problem as the result of rigidinstitutional expectations. Right-brained children are generally forgotten bycurrent school systems and their failure is rationalized and justified in thenormative system.

In a study by Edwards (1982), it was estimated that 80% of the generalpopulation in this country are left-brain dominant, which implies that only20% should have trouble learning in a school setting which is generallyperceived to be a left-brain institution.

Sperry (1974), Bogen (1975), and Bedford (1984) found that our schoolshave neglected the non-verbal form of intellect and that the entire student



body is being educated lopsidedly, using only the left hemisphere. Levy(1983) believes that modern technological civilizations depend primarily onleft-hemisphere functions and, therefore, people with potentially dominant,right-hemisphere capacities are victims of discrimination in modern advancedcultures.

Piatt (1979) studied the interrelationships between brain dominance andvarious traits of divergent youth with fifty-two male and female students inan alternative school for high school dropouts. The results indicated that over80% of the students in the sample were either right or mixed brain and only19% were left-brain dominant. The majority of the students were deemed tohave academic and avoidance problems and very few were referred forbehavioral problems. The results appear to support the assumption by manyresearchers in the field of brain processing that traditional left-brain teachingstrategies, such as lecturing, may be inadequate for students who are notleft-brain dominant (Scott, 1983; Hatcher, 1983; Levy, 1983).

School may be a frustrating, miserable place for a strongly right-brainedchild, particularly in mathematics. Such a child has difficulty gatheringinformation solely from verbal and symbolic sources, and is frustrated atbeing required to master bits and pieces when not been shown how they fitinto a whole. Their needed time for synthesis is continually interrupted.Writing successive steps in solving an arithmetic problem when the studentscan see the end result is nearly torture. Insights enable them to make intuitiveleaps that may lead to accusations of cheating.

Teachers chastise that "You don’t know it if you can’t explain it,"although as Wheatly (1977) claims, "Words are not the only medium ofknowing" (p. 37). As a result, strongly right-brain dominant childrenfrequently do not find school relevant or rewarding.Ferguson (1973) found that creativity scores invariably drop about 90%

between ages 5 and 7; by age 40, an individual is only about 2% as creativeas at age 5. This suggests that the almost total emphasis on logical, linearthought in our educational system may effectively suppress creativity.

Writers who criticize those who accuse schools of a left-brained orientationstress that schools require students to collect information and organize it andthat school subjects require integrative functioning for optimal learning(Levy, 1983). Meaningful reading (LB), for example, involves sensory,emotional, and experiential associations (RB) to achieve full appreciation ofconcepts. Problem solving in mathematics usually proceeds through threestages�playful attention to the problem in a non-directive way (RB), choiceof method and action in accord with that choice (LB), and reflection on theresults (RB). Writing (LB) is enhanced by experimental references (RB) andconceptualization skills (RB).

Speaking at the NCTM regional meeting in Albuquerque in October 1985,Joe Crosswhite stressed that the way in which students are tested inmathematics determines the way in which they will study. Most teacher-madeas well as standardized tests are designed to measure left-brain skills such as



simple recall, and knowledge level skills, the major determinant of achieve-ment and IQ. Thus, the creative student who perceives holistically rather thananalytically has little chance to achieve well on most of these tests.While there are certainly instances of students being expected to collect and

organize information, a great bulk of actual elementary and middle schoolmathematics is not characterized by this level of knowledge, but by simplerecognition and recall. At the "Seminar in Problem Solving" held at theUniveristy of Houston in 1986, Dr. Jesse Rudnick (presenter) stated that"90-95% of all activity in mathematics in grades 1-8 is concerned withcomputation," a left-brained function. Likewise, school subjects are amena-ble to integrative functioning; but is that a common approach to theinstruction? Not always; perhaps, not even frequently.

Developing Right-Brain Functions in Mathematics

As indicated in a previous section of this paper, several researchers haveexpressed serious concern that our schools have placed the bulk of theiremphasis upon left hemisphere functions, and that very little attention isgiven to right hemisphere activities. Levy (1983) has taken the position thatlearning is much more efficient and effective when both hemispheres areactively involved in the learning process. Normal brains are built to bechallenged, and the brain operates best when the cognitive processes are ofsufficient complexity to activate both hemispheres. Thus, simple, repetitive,and uninteresting problems would be poorly learned, with little benefit foreither hemisphere.

"Considerable evidence now suggests that the right hemisphere plays aspecial role in emotion and in general activation and arousal functions. Ifthis is so, if a student can be emotionally engaged aroused and alerted, bothsides of the brain will participate in the educational process" (Levy, 1983,p. 70).With this in mind, the following mathematical activities are proposed for

different grade levels and are designed to activate the right and lefthemispheres.

Elementary Grades

Young children have poor interhemispheric coordination, indeed the child atthe sensorimotor stage has very little left hemisphere activity, while the righthemisphere is quite dominant. At the preoperational stage, the righthemisphere is active, while the left hemisphere is developing, particularly inlanguage development. During the concrete operational stage, both hemi-spheres are active and the major emphasis in mathematics classrooms shouldbe coordinating the functions of the two hemispheres (Gazzaniga, 1974;Yacolev and Lacours, 1967; Harris, 1973; Bogen, 1975).The following activities are appropriate for any grade level, but particularly

for elementary level students:



(1) Supplying information in multiple formats including multisensory, experimen-tal opportunities, e.g., concrete manipulative exercises such as geometricshapes, fractional pies or squares, class discussion, videos, experimenting aswith probability concepts, and role playing. A good role playing vehicle wouldbe the following problem: If 8 people attend a party and each person shookhands with each other person exactly one time, how many total handshakeswould there be? Have the students guess, and record the guesses. Then have 8children stand and shake hands as specified. The total handshakes would be28, e.g., 7+6+5+4+3+2+1.

(2) Simple Addition�Use manipulatives and have students paste counters besidethe numbers. Some students need to see the whole pattern, so have them placethe counters where the answer will go.

(3) Story Problems�Have students draw pictures of what the problem is about�have them act out the problem; or have the students read the words, leavingout the numbers. Once they begin to visualize the problem, they should havelittle or no difficulty working it.

(4) Column Addition�Help students look for patterns within the problem such aspulling out 10 grouping numbers, multiplying numbers, or mixing the process.Ask students for other ways to solve the problem.

(5) Nine Tables�When presenting the 9’s multiplication tables, use a whole-to-part haptic (sense of touch) approach. Ex: 3 x 9�Have childen hold bothhands up. Tell them to count 3 fingers and fold the third finger down. Theywill have 2 fingers, a space, then seven more numbers, or an answer of 27.This works only with the 9’s tables. Practice by showing the children a fact byusing your hands and having them write the problem.

(6) Regrouping (Borrowing)�When subtracting, use color-code numbers, makingthe top number red and the bottom number green. Say, "Take the greennumber from the red number." By using color to organize the students’thinking, the students seem able to grasp the concept. For older students, havethem color or circle the numbers they must change.

(7) Dominoes and Dice�Have students memorize the dot pattern for eachnumeral. Once they have memorized the pattern, give them dominoes or pairsof dice and have them write the problems they see. Have them see how manydifferent problems they can make using only two dominoes or one pair ofdice.

(8) Numbers and Shapes�Use shapes to represent specific numbers. Have themvisualize the shape that represents the number. Then begin simple additionusing the shapes. (Vitale, 1985)

Elementary teachers should spend much more of their instructional time inthe use of concrete materials (manipulatives). For example, one of the mostdifficult concepts for elementary students to understand is division of wholenumbers. Rather than using rote learning techniques, as most of them do,they could use beans, M&Ms, popsickle sticks, poker chips, etc., to illustratethe concept, which is that division is the separation of a number of objects



into x equivalent sets of those objects. For example, 35 - 5: Using beans,the teacher would show the child that 35 is separated into 7 sets of 5;therefore, the answer is 7; seven what? Seven sets of 5. The teacher can thenlead the child to understand that division is successive substraction, theopposite of multiplication which is successive addition. This type of activityutilizes both hemispheres.

Middle School Grades (6, 7, 8)

These grades are supposed to be a transition period between the self-contained classroom of the elementary grades and highly specialized highschool grades. As such, the instruction should consist of use of manipulativesand lecture, which, however, is very seldom the case. Most instruction here isa carbon copy of the high school. Actually, manipulatives should be used atany grade level to promote effective learning.

Following are several concepts which are difficult to teach, but whoseunderstanding could be accomplished by more activation of the righthemisphere.

(1) Concept of denominator�Rather than being explained as "that thing on thebottom of a fraction," the perceptive teacher will illustrate the concept byusing fractional pies and squares, something the student can actually touchand manipulate. In this manner, the child can actually see that 1/6 is less than1/5, that 1/5 < 1/4, etc.

(2) Concept of Proportion�The following activity can easily be used to introducethe concept of proportion as well as enhancing a prior study of ratio. Thematerials include candy bars and play money. (This would be a good activityfor a special occasion when the students have earned the special treat of eatingthe candy bars after the activity.) Tell the students that the candy bars cost 2for $.25. Ask them to find the cost of 6 candy bars. If the students are givencandy bars and play money, they can quickly discover a solution to theproblem by placing one quarter beside each set of two candy bars until sixcandy bars have been accounted for. The solution is the total amount ofmoney. Obviously, this is a very simple type of problem but it clearlyillustrates the concept of proportion and the equivalence of ratios. This

activity can easily lend itself to a variety of expanded problems.

(3) Problem Solving (Puzzles)�Use puzzles, particularly of a spatial nature, whichrequire activities involving tessellations, pentominoes, tangrams, and soma cubes.

(4) Concept of Probability

(a) Create experiments which require students to make predictions and which usedifferent colored marbles, poker chips, etc. Have students put 10 red and 5white marbles in a container. Compute the theoretical probability ofwithdrawing a white marble, e.g., one out of three or five out of fifteen. Thenhave a student withdraw a marble from the container, record the result, andreplace the marble. Do this 30 times and compare results with theoreticalprobability. Have class discuss the results. -


126Left Brain, Right Brain

(b) Finding Probability of an event with repeated trials�A probability tree can beused for this activity. Example: A coin is tossed three times. Find theprobability of getting heads, heads, tails.There are 8 possible outcomes. One is heads, heads, tails, P(H, H, T) = 1/8.

A coin is tossed three times. Find each probability:1. P (exactly 2 heads)2. P (at least 1 head)3. P (3 heads)4. P (at least 2 tails)

T H

(H,T,T» (T,H.H»

(5) Operations on Integers�This exercise is designed to teach addition, subtrac-tion, multiplication, and division of integers (directed numbers) by the use of amodel in which the concept of positive and negative "charges" is used. Theteacher would use an empty field represented by a circle. Pluses (+) and minuses( �) are made of construction paper and are moved in and out of the circular fieldto illustrate the four operations. The reader is referred to an article by StanleyCotter which appeared in the May 1969 Arithmetic Teacher (pp. 349-53). Cottergives an excellent discussion as well as numerous illustrations.

High School Grades

For the high school grades, the author offers three examples of activating theright hemisphere while teaching mathematics. The literature is replete withexamples concerning algebra, analysis, probability, etc.

(1) Algebra�The teacher should cut from construction paper, several squares (allthe same size), several strips, and several unit squares. These materials will beused to factor trinomials or to construct trinomials from factors. An examplewould be: Factor the trinomial by using the squares, strips and unit squares.The physical representation of the trinomial is given in Figure 1. Thefactorization is illustrated in Figure 2.

X2

2x

x

+

I

3x

FIGURE 1

X X x

+

1

D

2

x + 1

x + 2

FIGURE 2



The factors, as one may observe are (x + 1) (x + 2) which are the dimensionsof the figure constructed by use of the concrete materials. Much more in-depthinformation may be obtained by reading "Finding Factors Physically" byChristian R. Hirsch, which appeared in the May 1982 Mathematics Teacher.

Activities such as this enable students to visualize concretely a ratherabstract principle, and in addition activate the right hemisphere as well as theleft brain, thus enabling the "whole" brain to become active. Such activitiesenable students to learn more efficiently and effectively.

(2) Functions�Teachers of mathematics should always strive to enable students to

visualize functions. For example, students need to understand why an equationof the form ax + by = c is designated as a "linear" equation and anequation of the form ax2 + bx + c = y is called "quadratic." These two

functions, one "linear," and one "quadratic," are geometric, spatial func-tions; being able to visualize a straight line for ax + by = c and a curve for y= ax2 + bx + c is essential to achieve understanding of these concepts. Somechildren learn best when they are taught first the geometric spatial relationshipand then are taught these geometric forms in a symbolic equation (Levy,1983).

3. Rotations and Reflections�Teachers have a great deal of difficulty teaching thestructure of the various number systems, e.g., natural numbers, wholenumbers, integers, etc., with understanding. Students very seldom understandthe relationship of the properties to the operations, e.g., what the propertiesallow one to do with the operations and, in general, what makes the systemwork. One strategy used by the authors is to use the rigid rotations andreflections of the equilateral triangle to generate a noncommutative system. Theelements of the system are created by physically reflecting and rotating thetriangle, such as:

Using a larger triangle to represent standard position, we can let our first threeelements be rotations of the triangle in a counterclockwise direction. Thus, arotation through 120° will be defined to be element

D =123

312



That is, a rotation through 120° takes the vertex numbered 1 into standardposition 3, 2 into 1, and 3 into 2. Two 120° rotations, or a rotation through240° will be called element

/I 2 3’E; [2 3 ^Likewise, three 120° rotations, or a rotation through 360° will be the element

which we shall denote by

�; (’ 2 3)\1 2 3/The remaining three elements will be reflections of the triangle onto itself; seeFigure II. The reader can visualize axis la as a rod. The reflection of thetriangle across the rod will be our fourth element

A.f1 2 3)M 3 2/Continuing, we have the reflection of the triangle across the axis Ib which we

will call element y

B./1 ’ 3)\3 2 1 /and the reflection across axis 3c, element

c.(1 2 3)\2 1 3/Thus, we have the set of elements |A, B, C, D, E, I].

2

FIGURE 2



For any two elements x and y of the set, we may define the binary operation oon this set, x o y, to the first x followed by y. For example, in order to find Ao C, one would have to reflect on rod la (see Figure II) and then reflect onrod 3c from standard position. On paper we have

A 0 C ^ 2 3} � f1 2 3}’ f1 2 3}- E- ^1 3 2) \2 1 3/ \2 3 I/which is the rotation through 240°. This is in effect permutation multiplication.

The key to success in effectively teaching this concept is the physicalmanipulations of the triangle to generate the elements of the system and tooperate on the elements. The same thing can be accomplished with rigidrotations of the square and pentagon (Creswell & Wiscamb, 1970).

It is imperative that we as teachers teach in such a manner that will enableour students to learn with maximum efficiency and effectiveness. It is feltthat using activities similar to those listed above will activate bothhemispheres of the brain which will help achieve this goal.

Whole-Brain Learning

Those who encourage right-brain activities in schools do not generally intendthat such be used exclusively (even for some children), but are recommendingtheir addition to a curriculum already filled with left-brained activities(Hatcher, 1983; Kail, 1981). There are two purposes. First, such activitieswould bring out the abilities of many of those who are not succeeding in thecurrent system because their strengths lie in different areas. Second, thepresence of stimulation to both hemispheres would encourage all children todevelop integrative, collaborative information processing by both hemi-spheres.Levy (1983) says that attention to brain dominance should not mean that,

for a particular child, one hemisphere should be the object of education, butthat "gateways into whole-brain learning may differ from different children"(P. 70).Hart (1975) claims that brain-compatible approaches could increase

achievement in schools by 100-200%. As indicated above, there are amultitude of measures that can encourage right-brain activity in schools.Provision of time in a reduced stress atmosphere allows the right hemisphereto synthesize (a function that is often unable to occur during prominentleft-brain demands). Emotional closeness within a classroom and an enrichedenvironment are factors that increase right-hemispheric productivity andintegrative functioning (actually stimulate myelinatin which physically en-hances integrative capabilities). Allowing the expression of a response by aparagraph or a chart, diagram or poster, or providing opportunity forphysical or oral demonstration of knowledge allows for varying methods ofresponse.



"Research results indicate that normal brains are built to be challenged,that they only operate at optimal levels when cognitive processing require-ments are of sufficient complexity to activate both hemispheres and provide amutual facilitation as they integrate their simultaneous activities. Education-ally, this means that simple, repetitive, and uninteresting problems (such asmost mathematical computations) would be poorly learned, with little benefitfor either hemisphere" (Levy, 1983, p. 70).

Is Attention to Brain Research a Fad?

The brain is the central organ for learning, which makes it fundamental toeducation. It controls emotion and behavior, which are integrally supportiveto the educational process. Thus, knowledge about the brain must take anenduring place in the attention of educators. Only recently has enoughresearch been done in this area to justify application. It is crucial thateducators become knowledgable in areas such as this and build a theoreticaland empirical basis for educational practice.

References

Bedford, C. (1984). Why are we learning this? Mathematics Teacher, 77(4), 258-263.Bogen, J. (1975). Some educational aspects of hemispheric specialization. UCLA

Educator, 77, 24-32.Cotter, S. (1969). Changed particles: A model for teaching operations with directed

numbers. The Arithmetic Teacher, 76(9), 349-53.Creswell, J. L., and M. Wiscamb, (1970). New strategies for teaching properties ofnumber systems. School Science and Mathematics, 70(7), 635-645.

Edwards, C. H. (1982). Brain function: Implications for schooling. ContemporaryEducation, 53(2), 58-60.

Ferguson, M. (1973). The brain revolution. New York: Taplinger.Gaylean, B. (1981). The brain, intelligence and education. Roper Review,Gazzaniga, M. S. (1974). Cerebral dominance viewed as a decision system. In S. J.Dimond and J. G. Beaumont (Eds.), Hemisphre function in the human brain. NewYork: Wiley & Sons.

Harris, L. J. (1973). Neurophysiological factors in spatial development. ERIC ED 036163.

Hart, L. (1975). How the brain works. New York: Basic Books.Hatcher, M. (1983). Whole-brain learning. School Administrator, 40(5), 8-11.Hirsch, C. R. (1982). Finding factors physically. Mathematics Teacher, 75(5), 388-393,

419-422.Johnson, V. R. (1982). Myeline and maturation: A fresh look at Piaget. Science

Teacher, 49(3), 41-44, 49.Kail, R. A. (1981). The battle within your brain. Success, Oct., 20-23, 48-49.Levy, J. (1983). Research synthesis on right and left hemispheres: We think with both

sides of the brain. Educational Leadership, 40(4), 66-71.Piatt, J. (1979). Hemisphericity and divergent youth: A study of right-brained

students, their school problems and personality traits. Unpublished doctoraldissertation. International Doctoral Dissertation, 40, Order No. 7919011.

Power, J. (1984). Gray matters. NEA Today, 2(8), 4, 5.



Scott, P. B. (1983). A survey of perceived use of mathematics materials by elementaryteachers in a large metropolitan school district. School Science and Mathematics,61-68.

Sperry, R. W. (1974). Lateral specialization in surgically separated hemispheres. InSchmitt and Worder (Eds.), The Neurosciences Third Study Program (5-19).Cambridge, Massachusetts: MIT Press.

Springer, S. P., and G. Deutsch (1981). Left brain, right brain. San Francisco: Free-man.

Vitale, B. (1985). Unicorns are real. California: Jalmar.Weibe, J. H. (1981). The use of manipulative materials in first grade mathematics: A

preliminary investigation. School Science and Mathematics, 388-390.Wheatley, G. H. (1977). The right hemisphere’s role in problem solving. The

Arithmetic Teacher, 24{\}, 36-38.Wittrock, M. C. (1981). Educational implications of recent brain research. Educational

Leadership, 39(1), 21-15.Yacolev, P., and A. R. Lacours (1967). The myelogenatic cycles of regional

development of the brain. In A Minowski (Ed.) Regional development of the brainin early life: Symposium. Philadelphia: Davis.

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implications of right/left brain research for mathematics educators

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