Counterintuitive Instances Encourage Mathematical ThinkingAuthor(s): MARSHALL GORDONSource: The Mathematics Teacher, Vol. 84, No. 7 (OCTOBER 1991), pp. 511-515Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27967268 .

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Counterintuitive Instances

Encourage Mathematical

Thinking By MARSHALL GORDON

Intuition,

experience, and reason are the

primary modalities through which hu man beings make sense of their environ ment and gain knowledge. Our intuition, which senses a situation immediately, has considerable weight, of course, with regard to what we believe (Fishbein 1979) and so deserves the attention of teachers and text

book writers involved with mathematics ed ucation. The use of intuition in instruction includes presenting mathematics examples that are counterintuitive. For not only do instances that run counter to intuition gain students' attention because of the disequilib rium experienced when what had been

imagined to be true turns out not to be so, but such examples also help students chal

lenge habits of thought and practices, thus

leading to their becoming better thinkers

(Marzano et al. 1988, 128). By presenting students mathematical moments that chal

lenge common sense and common practice, the teacher gives them the opportunity to

gain a greater appreciation of the need for

exploration, reflection, and reasoning. Counterintuitive moments can be found

in all areas of mathematics. Classic illustra tions occur where the infinite and the finite come together, as in the demonstration that 1 = 0.9 and when y

= II is revolved around the x-axis over the interval [1, o?) to produce a solid having finite volume and infinite sur

face area (McKim 1981). Another topic in which surprises are often found is prob ability, as in the famous birthday prob

Marshall Gordon teaches at the Park School, Brook

landville, MD 21022. He is interested in the construction

of knowledge as it comes to be in students' minds and as

it appears in textbooks.

lem wherein it is determined that among twenty-three randomly chosen people, the

probability is greater than 0.50 that two of them will have the same birthday (Gold

berg 1960).

Counterintuitive moments create disequilibrium and foster exploration.

The following examples are offered for

enlivening students' mathematics experi ence by challenging their intuition:

1. Secondary school students are taught volume formulas for a number of three dimensional objects, factoring, and the com

mutative and associative properties of mul

tiplication over the natural numbers. The counterintuitive solution to the following problem can help them appreciate that the interface of algebraic and geometric ele

ments can enrich their mathematical under

standing. The problem is to determine the change

in volume of a 12 x 18 x 24 (width by height by length) fish tank (rectangular parallel epiped) when one of the dimensions is dou

bled. Students' intuition generally leads them

to claim initially that doubling the length (which has the greatest magnitude) has the greatest effect on the volume, whereas dou

bling the width has the least. They appear concerned when they determine that the vol umes resulting from doubling any of the

dimensions are all the same. Suggesting that they focus on the elements in the prod

October 1991 511

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uct they used to determine the volumes

helps them to note that 2 can be factored out

from each of the three products, so doubling any one dimension doubles the volume

regardless of the relative sizes of each di

mension:

(2 12) 18 X 24 = 12 x (2 x 18) 24 = 12 18 (2 X 24) = 2 x (12 x 18 x 24).

This numerical explanation seems quite convincing, but a more complete under

standing comes from looking at how the

shape of the fish tank is affected in each

case; namely, by discussing what doubling each of the dimensions actually does in fig ure 1:

a) Doubling the height in effect creates a

second tank of the same dimensions on top of

the first.

b) Doubling the width in effect puts a second tank next to the first.

c) Doubling the length in effect puts a second tank behind the first.

2. Another geometric counterintuitive instance that stirs mathematical thought oc

curs when students are asked to determine

the distance between the earth's surface and a concentric circle whose circumference is 30

feet more than that of the earth, which is

approximately 25 000 miles. Students usually jump to the conclusion

that the difference is very small, given that a 30-foot addition to the 132 000 000-foot circumference is indeed negligible. They are

quite bewildered when they determine that

the difference is very close to 5 feet! The surprising calculation offers an ex

cellent opportunity to discuss the virtue of

the problem-solving strategy of abstracting a problem so as to understand its structure

Uc7 Fig. 1. Doubling different dimensions of a fish tank

512_:_

beyond the individual details. By realizing that the radius (r) and circumference (C) are

linearly related, students can determine that a unit increase in r generates a 2 unit

increase in C. Thus, if the circumference of the earth is C =

2irr, then the increment, , in the radius results in the greater circum ference

C + 30 = 27r(r+ ),

and thus 2 = 30, yielding

? 4.77 feet.

Believing is seeing.

Students' confusion is grounded in their

trying to bring a single perspective to two

contexts; namely, relative to the earth's ra

dius (approximately 4 000 miles), an in crease in approximately 5 feet is indeed neg

ligible, whereas relative to the height of an

individual, the increase is significant. Thus, this instance offers an opportunity for dis

cussing that one's perspective has implica tions for what is believable?that is, believ

ing is seeing. Such a discussion could

include the surprise and antagonism that

may accompany the introduction of a new

number or geometric system that includes

aspects contrary to the way we have come to

understand the nature of number or space.

3. In solving systems of equations, stu

dents are taught to eliminate variables and

equations to secure one equation in one un

known with the goal of determining whether a common solution exists. This technique is, of course, an instance of the generalization that simplifying situations helps one gain

insight into the underlying relationships. So

students are initially perplexed when the

suggestion is made that they introduce

greater complexity into a problem to gain

greater understanding. This approach has

validity in various mathematical instances, as when we add the same quantity to, or

subtract it from, an equation or multiply by a nonobvious expression having the value of one. The following is another instance where

understanding can be gained by introducing

greater complexity.

_ Mathematics Teacher

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After solving systems of equations in the

usual manner, students can consider adding

equations and imagine how the characteris

tics of the resulting equation compare with

those of the original equations. If this expe rience of adding equations leads students

away from determining what is involved in

solving systems of linear equations, then

they may not have obtained any mathemat

ical understanding of value from this inves

tigation.

Abstraction reveals the problem's structure.

For pairs of linear equations in which

lines are parallel (vertically or horizontally) or intersecting (with both having either pos itive or negative slopes), the sum of any pair creates a third equation whose "genealogical roots" are clear in that the resulting line can

be seen to share significant characteristics

with the two generating lines. That is, given lines y

= a and y =

b, then the "summed"

line, 2y = a + b, yields

a +

Similarities can also be found for two verti

cal lines: the summed line is halfway be

tween the original lines and has the same

orientation to the axes.

In the example of two oblique lines, y -

ax + b and y = cx + d, the summed line is

easily found to be

(a + c\ , b + d

where the arithmetic means for the slope and the y-intercept seemingly represent a

balance in the "offspring" of the "parent" lines. Here, reasoning by analogy can be

seen to be potentially valuable in promoting mathematical thought. And this perspective takes on a special meaning when students

discover that all three lines contain the

same point of intersection:

id ? b ad ? bc\ \a ? c9 a - c /

The grapher of a pair of oblique lines is

advised to begin with both slopes, whether

positive or negative, because the summed

equation that represents the average physi

cally appears between the two given lines.

(That slopes with different signs create a

different image will be discussed shortly.) The observation that the new line has the

average slope and y-intercept leads a num

ber of students immediately to assume that

two of the angles created by the two original lines are bisected by the so-called offspring line. They are surprised to find that, in gen

eral, angle bisection does not occur. This

counterintuitive moment serves as a won

derful jumping-off point for considering the

distinction between angular and linear mea

sure.

When the two original lines have slopes with different signs, the summed line does

not fall between them. This inconsistency can promote a discussion about the complex

ity of unifying two systems, as in seeking to

quantify space. For example, in considering Cartesian two-space, students may wish to

reflect on the fact that as k increases, both

y = kx and y

- -kx begin to move toward

becoming the line = 0.

4. Students studying an area of mathe

matics are subconsciously developing an in

formed intuition in the process. So when a

counterintuitive mathematical instance is

presented, the disequilibrium can be consid

erable, and the students' surprise can moti

vate them to uncover why things are not

what they had assumed.

Introducing complexity can aid understanding.

In the following situation, we assume

that students have learned about matrices,

perhaps in the study of transformational ge

ometry, probability, networks, or economics.

And so when they are introduced to abstract

considerations, they may feel quite comfort

able because of their exposure to matrix op erations. Thus, it is natural in comparing

"ordinary" algebra with matrix algebra

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that, since ab = 0 in ordinary algebra im

plies that either a = 0 or b = 0, students

assume that if A = 0, where A and are

matrices (e.g., square matrices of order 2), then A = 0 or = 0. Presenting them with the example

1 4"

14 " 3 3J

and

AB = 2 6 5 15

0 0 0

introduces them to the fact that matrix al

gebra and "ordinary" algebra do differ. The

question that arises after confusion subsides

is, In general, do nonzero matrices A and exist such that AB = 0? And if students have made the passage into the formal alge bra, one might well ask, Do nonzero forms of A exist such that A is a nonzero square root of 0?that is, where A A = 0? Both answers can be affirmative.

If we let

a b c d

and

= e gh\>

then AB = 0 implies that?

(1) ae + bg =

0, (2) af+bh = 0, (3) ce + dg

= 0,

(4) cf+dh = 0. From (1) and (2), ae + bg

= af + bh, which

can be restructured as

a(e

For g h and n = b(h-g).

e-f v

h-g'

it follows that b = ka, and similarly for (3) and (4), d = kc. From setting (1) = (3) and (2) = (4), it can be shown that for b d, g =

je and h = jf, where

J b-d'

Thus, when h g and b d, A and assume the forms

a ka c kc.

e r jejf.

respectively. However, the relationship of A and is

even more structurally intimate than now

appears. For b = ka, d = kc, g

= je, and h =

jf, we can rewrite (l)-(4) as follows:

(5) ae(l + kj) = 0 (6) af(l + kj) = 0 (7) ce(l + kj) = 0 (8) cf(l + kj) = 0

If 1 + kj 0, then ae = af

= ce = cf which

implies that either A = 0 or = 0. So to

identify the nonzero divisors of 0, we must have 1 + kj

= 0. For k 0J = -Ilk. Hence,

we find that a ka] A = c kc

e

e_ k

r

k.

and A = 0. To include determining the square roots

of the zero matrix, when A = B, we have

that

(9) a = e,

(10)

(11)

and

(12)

ka ? f, e

kc = ?

which implies that

a ka a

~k~a.

and A A = 0. The foregoing counterintuitive instances

can be used to introduce topics, stimulate

deeper development of a topic, and corrobo rate students' understanding. Set theory (Love 1989) and number theory (Brown and

Walter 1983) are other areas of mathematics that furnish wonderful counterintuitive in stances for students and teachers to explore.

514 Mathematics Teacher

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BIBLIOGRAPHY Brown, Steven L, and Marion Walter. The Art of Prob

lem Posing. Hillsdale, N.J.: Lawrence Erlbaum Asso

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Davis, P. J. The Mathematics of Matrices. Waltham, Mass.: Blaisdell, 1965.

Fishbein, E. "Intuition and Mathematical Education."

In Some Theoretical Issues in Mathematics Educa

tion: Papers from a Research Presession, edited by Richard Lesh and Walter Secada. Columbus, Ohio:

ERIC Clearinghouse for Science, Mathematics, and

Environmental Education, Ohio State University, 1979.

Francis, Richard L. "Word Problems: Abundant and

Deficient Data." Mathematics Teacher 71 (January

1978):6-10.

Goldberg, Samuel. Introduction to Probability. Engle wood Cliffs, N.J.: Prentice-Hall, 1960.

Love, William, P. "Infinity: The Twilight Zone of Math

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92.

McKim, James. "Problem of Galaxia: Infinite Area ver

sus Finite Volume." Mathematics Teacher 74 (April 1981):294-96.

Marzano, Robert J., Ronald S. Brandt, Carolyn Sue

Hughes, Beaufly Jones, Barbara Z. Presseisen, Stuart

C. Rankin, and Charles Suhor. Dimensions of Think

ing: A Framework for Curriculum and Instruction.

Alexandria, Va.: Association for Supervision and

Curriculum Development, 1988.

School Mathematics Study Group. Mathematics for

High School: Introduction to Matrix Algebra. Rev. ed.

New Haven, Conn.: Yale University Press, 1961. W

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