counterintuitive instances encourage mathematical thinking
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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
October 1991 513
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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
ciates, 1983.
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
ematics." Mathematics Teacher 82 (April 1989):284
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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