Visual Thinking, Algebraic Thinking, and a Full Unit-Circle DiagramAuthor(s): JONATHAN SHEARSource: The Mathematics Teacher, Vol. 78, No. 7 (OCTOBER 1985), pp. 518-522Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27964605 .

Accessed: 13/07/2014 06:37

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Mathematics Teacher.

http://www.jstor.org

This content downloaded from 132.203.227.63 on Sun, 13 Jul 2014 06:37:06 AMAll use subject to JSTOR Terms and Conditions

Visual Thinking, Algebraic

Thinking, and a Full

Unit-Circle Diagram By JONATHAN SHEAR, Bellingbrook School, Petersburg, VA 23805

Tlhe Chinese epigram A Picture Is Worth

a Thousand Words vividly reflects the

different, often complementary roles of two

cognitive styles with which we are all fa miliar. One of these is visual?spatial and holistic in nature; the other is verbal? linear or sequential in nature. It is well ac

cepted among psychologists and re searchers in creativity that creative think

ing is most effective when both of these

styles of functioning are active in a coordi nated manner. Since Jacques Hadamard's

ground-breaking work The Psychology of In vention in the Mathematical Field (1945), it has been widely recognized that mathemat ical thinking is no exception to this general rule.

Two examples of autobiographical com ments by eminent mathematicians will

readily illustrate the nature of visually ori ented holistic thinking and contrast it with

sequential, verbal types of thought. In his famous Scientific American article of 1948, Jules Henri Poincar? (1968, 17) described his own creative thought processes as un

folding in visual terms, through perceiving mathematical entities ... whose elements are harmoni

ously disposed so that the mind without effort can em brace the totality while realizing their details.

Here he would "see at a glance the reason

ing as a whole" (p. 17) rather than

sequentially?despite the fact that his in

sights will later be expressed in the verbal, formulaic, logically sequential form charac teristic of finished mathematical works.

Einstein, too, wrote that his creative

thought was visual and nonverbal in nature

(1979, 35-36):

Words or ... language, as they are written or spoken, do not seem to play any role in my mechanism of

thought. The psychical entities which seem to serve as

elements in thought are certain signs and more or less clear images_

The above-mentioned elements are, in my case, of visual and some of muscular type. Conventional words or other signs have to be sought for laboriously only in a secondary stage.

Visual, spatial thinking, where compo nents and relationships are perceived at

once, all together, is of course not confined to geniuses. We are all familiar with it, and it plays an important role in mathematical

thinking in general, just as it does in other fields. Despite this fact, many students, es

pecially at the secondary school level, tend to regard mathematical thinking as being largely if not almost entirely verbal in nature. This view is due partly to the

highly algebraic, sequential form in which mathematical work is typically expressed, partly to the fact that mathematics is

taught largely in terms of such formulaic

expressions, and partly to the fact that stu dents frequently conceive their work to be a process of solving formulaic problems and

producing correct "

answers." The way

mathematics is taught and learned thus often has very little to do with the way mathematics is created by mathematicians. As a result students often fail to develop the visual, nonverbal component of their mathematical thinking. This not only can

hamper their grasp of classroom material but also is likely to diminish their appreci ation, both aesthetic and intellectual, of mathematics as a whole.

Coordinate geometry, itself an unus

ually clear expression of the integration of

visual-spatial and verbal-algebraic think

ing, provides an excellent opportunity to

help the student to develop both of these

very different styles of thinking in a coher ent way and to see how each can powerfully

518 Mathematics Teacher

This content downloaded from 132.203.227.63 on Sun, 13 Jul 2014 06:37:06 AMAll use subject to JSTOR Terms and Conditions

enhance the other. Theorems that are diffi cult to prove algebraically will often be

easy to understand and prove geo metrically, and vice versa. Furthermore, ge ometric figures can often be used not only to help the student learn the relevant alge braic relations and operations but also to show how visually oriented thinking can

provide information and insights that may be neither obvious nor easy to remember

algebraically. The study of trigonometric functions in

terms of the unit circle offers a good exam

ple of how geometric figures can be used in this way. As we shall see, a single rather

simple picture can be constructed that will at once (a) show the six basic trigonometric functions, (6) display the three basic Pytha gorean relationships with their six major (and other minor) variants, (c) display the relations of functions, angles, cofunctions, and complements, (d) provide the basis for

simple, intuitively obvious proofs for the

preceding and other (e.g., the reciprocal) trigonometric relations, and (e) clarify and make use of the names of the trigonometric functions (tangent, secant, etc.).

Until the 1960s the teaching of introduc

tory trigonometry was generally based on

right triangles, with the trigonometric func tions of an acute angle 0 defined in terms of the hypotenuse (A) and legs (lx and l2) of a right triangle. Since that time the use of the polar, or unit-circle, approach, at once

simpler and more general, has become the dominant method of teaching trigonometric relations. Here the trigonometric functions of an angle 0 are defined in terms of the intersection (x, y) of the unit circle and the

ray from the origin with inclination 0. Sin 0 and cos 0 are defined as the and y coordi

nates, respectively, of this intersection, as in figure 1.

Tan 0 is defined as

sin 0 y cos 0

9

and then cot, tan, and sec are defined as inverse functions, with cot 0 = l/tan 0, sec 0 = 1/cos 0, and esc 0 = 1/sin 0. The stu dent is shown, using figure 1, that sin2 0 +

Fig. 2

cos2 0 = 1. Other relationships based on the

Pythagorean theorem and the cofunction formulas are then typically developed alge braically. The student is then expected to memorize and master the use of the various formulas.

This standard approach can be signifi cantly enhanced by the use of a more com

plete unit-circle diagram, one that simulta

neously increases the student's understand

ing and relieves the burden on (and uncer tainties of) his or her memory. The diagram can be built up, step by step, as follows.

First, the segment of the tangent lying between the point (x, y) of tangency and the x-axis is drawn and labeled "tan 0." See

figure 2. Students should readily see that

cos??12

October 1985 519

This content downloaded from 132.203.227.63 on Sun, 13 Jul 2014 06:37:06 AMAll use subject to JSTOR Terms and Conditions

are similar right triangles, that their corre

sponding sides are proportional, and that if we reference the hypotenuses, the legs op

posite to 0, and the legs adjacent to 0 by ,

Zj, and Z2, respectively, we have

sin 0 , tan 0 -2

= r

= ?:? = tan 0, cos 2 1

thus showing that the length of the segment of the tangent line labeled "tan 0" fulfills the usual definition for tan 0. Similarly, if we label the segment of the secant line

lying on the x-axis between the origin and the tangent line "sec 0" as shown in figure 3, we see that for the section of the geo metric secant labeled "sec 0,"

1 h sec 0 -= ? =-= sec 0.

cos 0 Zr 1

The second major Pythagorean relation,

namely,

sec2 0 = tan2 0 + 1,

will now also be readily seen and accepted. Figure 3, in addition to being useful

mathematically, displays the geometric sig nificance of the terms "tangent" and "secant." The original meanings of "co

tangent" and "cosecant" now provide the

guide for a more complete, and much more

useful, diagram. Historically, the cosine,

tangent, and cosecant of an angle were de fined as the sine, tangent, and secant, re

spectively, of the complement of the angle. Modern trigonometry no longer uses these

definitions, but it preserves the relevant al

gebraic relations. In triangle-based trig onometry two of these cofunction relations

(sine-cosine and tangent-cotangent) are still

visually apparent. None of them, however, is at all obvious in unit-circle trigonometry as it is usually taught. But these relations can and ought to be seen easily, as a "full" unit-circle diagram makes clear.

It should first be noted that the acute

angle adjacent to 0 is its complement, /2

? 0. The segment of the tangent at (jc, y) between (jc, y) and the j'-axis is drawn in, and it is noted that this segment stands to

the angle /2 ?

0, the complement of 0, as

the tan 0 segment stands to 0. The new seg ment is then labeled "cot 0" in conform

ity both with the historical definition of "cot 0" and with the algebraic relation cot 0 = tan ( /2

? 0). The newly intersected

segment of the ^-axis, which stands to

( /2 ?

0) as the sec 0 segment stands to 0, is now labeled

" esc 0.

" This result yields the

"full" unit circle in figure 4. The use of pairs of similar right tri

angles now quickly shows that

520

Fig. 3 Fig. 4. "

Full "

unit-circle diagram for angle 0

-Mathematics Teacher

This content downloaded from 132.203.227.63 on Sun, 13 Jul 2014 06:37:06 AMAll use subject to JSTOR Terms and Conditions

or, equivalently, cos 0/sin 0, and that

esc 0 A 1

The lengths of the segments thus fulfill the usual algebraic definitions of cot 0 and esc 0 given earlier, and the segments are

therefore named appropriately in algebraic as well as historical terms. The third major Pythagorean relation, csc20 = cot20 + 1, and its variants are now also readily appar ent from the diagram.1

Most students will readily see that the cot 0 and esc 0 segments of figure 4 stand to the central angle complement of 0 as the tan 0 and sec 0 segments, respectively, stand to 0. Figure 5, generated by reflecting figure 4 across the line y = je, can, if neces

sary, be used to make it clear to the rest. A few labels and one final supplementary line added to figure 5(6) makes the complemen tary roles of sine and cosine clear as well.

Once these drawings and demon strations have been made during a class ses

sion, and are reviewed occasionally, even

less able students can remember and repro duce figure 4 (referred to in class as "the full unit-circle diagram for angle 0") and use it easily to grasp correctly and generate in writing the basic Pythagorean and co

function relationships. It is worth noting that the graphic display (and simple geo metric proofs) of the cofunction relation

1. By referring to the figure while working on problems, stu dents also discover and use other, less obvious, Pythagorean relationships. For example, one of the problems from the

chapter test (in the teacher's manual) for chapter 7, "Circle

Trigonometry," in Richard Brown and David Robbins's Ad vanced Mathematics (Boston: Houghton Mifflin Co., 1978) asks the student to show the identity

(sec - cos )2 = tan2 - sin2 .

The solution Brown and Robbins propose is

(sec - cos )2 = sec2 - 2 sec cos + cos2 0

= sec2 0 - 2 + cos2 0 = (sec2

- 1)

- (1

- cos2 0) = tan2 0 - sin2 0.

Some students have difficulty grasping and reproducing deri vations such as these properly?much less arriving at them on their own. Yet these same students can often discover the relevant identities simply by "reading them off" the figure? as happened in the case at point (using th? small triangle having tan 0 as the hypotenuse and sin 0 and (sec 0 ? cos 0) as the legs). Furthermore, all the students in the class preferred this graphic method once it was pointed out.

(C) Fig. 5

ships is particularly helpful to those many students who find the usual algebraic dem onstrations less than intuitively obvious and who consequently may fail to remem

ber the relationships properly. Fur

thermore, understanding the geometric sig

October 1985 521

This content downloaded from 132.203.227.63 on Sun, 13 Jul 2014 06:37:06 AMAll use subject to JSTOR Terms and Conditions

nificance of the trigonometric terms

"tangent" and "secant" gives students a

greater sense of the unity of mathematics and also helps them to remember and repro duce the figure correctly; proper under

standing of the common significance of the

prefix "co-" in "complementary," "cosine,"

"cotangent," and "cosecant" is similarly helpful for the student's correct memory, understanding, and use. It may also be worth noting that the expressions of fasci nation and even delight that can cross stu dents' faces when they first recognize how much information is expressed in this

simple schematic figure can be a pleasure to a hardworking teacher, and may even be a sign that the student is gaining some sense of that elegance of thought and ex

pression which, in the words of Poincar?, "all true mathematicians know."

REFERENCES

Einstein, Albert. Ideas and Opinions. New York: Dell

Publishing Co., 1979.

Hadamard, Jacques. An Essay on the Psychology of In vention in the Mathematical Field. New York: Dover

Publications, 1945.

Poincar?, Jules Henri. "Mathematical Creation." In Mathematics in the Modern World, Readings from Sci

entific American, edited by Morris Kline. Reprint 1948. San Francisco: W. H. Freeman & Co., 1968.

101 PUZZLE PROBLEMS With Solutions by

Bates-Smith Develops sound problem-solving technique Excellent math enrichment material

Good for "day before vacation use"

Stimulating extra credit problems Suitable for use with computer

Sample SAT Math Problems & Math fallacies

Intro, number theory, historical notes

at $8.00 (price includes postage) Send check or money order to:

NATHANIEL B. BATES 277 Nashoba Rd., Concord, MA 01742

This geometry text is just

like the teachers who use it.

Successful. Geometry for Enjoyment and

Challenge is a best seller, for good reason.

It really works. It puts geometry in such

understandable terms that teachers find students are actually able to read and un derstand basic concepts and terms presented in the book.

This frees the teacher to spend class time doing valuable problem-solving activities and discussing broader math concepts.

McDouoal. P.O. Box 1667,

As one teacher said, "My students are now more successful (thanks to

Geometry for Enjoyment and Challenge)... students who previously would be struggling are succeeding... average students enjoy being able to excel... excellent students enjoy being chal lenged by more than drill!" We invite you to see for

yourself. Ask for your com plimentary copy today.

Just phone toll-free 1-800-323-4068. In Illinois, call collect

Littell & Company 312-967-0900. Evanston, IL 60204

GEOMETRY

G

522 Mathematics Teacher

This content downloaded from 132.203.227.63 on Sun, 13 Jul 2014 06:37:06 AMAll use subject to JSTOR Terms and Conditions