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Instead of asking students "Show that there is no...." I
generally ask my students "Find a..." without giving any
additional information. The reader may consider I lied to
them... I don't. With the task formulated affirmatively, the
solver will try to accomplish the task and only after same
attempts (sometimes many attempts or even never...) he will
suspect that the task is "very hard", or almost
"impossible". Then, he is in front of a metamathematical
decision: to keep on trying to do what he was asked to do or
to be "insolent" and change direction trying to prove that
he had been asked to do something that is not possibly done.
From an educational point of view, I think it is very
important to have mathematics students discuss the role of
impossible things in mathematics.
First of all, it is necessary to
acknowledge that there exist impossible "things" in
mathematics. Some of the questions that lead to such
"things" were asked from antiquity (i.e. Trisecting an
angle, Doubling a cube, Squaring a circle - all of the above
with straightedge and compass alone) and the attempts to
give them an answer lead to the development of very
important branches of the mathematical knowledge. Second,
the fact that there are impossible things in mathematics is
quite surprising. The analysis of results about
impossibility may lead students to a more real perspective
of the subject. Third, the discussion of results of the form
"It is impossible to..." may constitute a good opportunity
to clarify the distinction between unsolved problems and
unsolvable problems. As Davis said, "There seems to be a
time element at work in such [impossibility]
statements. Actuality is here, actuality is now, it is
complete; an impossibility seems to bargain with an
uncommitted future." (Davis, 1986, p.67)
In the frame of a course for pre service
secondary school mathematics teachers, the classic proof
that there exist at most five regular solids was presented.
Later on, during the same lesson, we showed Kepler's version
of the Solar System and we discussed the connections between
the Platonic Solids and the orbits of the planets. The
following dialog occurred:
S [student]: At Kepler's time only some
of the planets were
known. But nowadays, we know that there are nine
planets.
T [teacher]: Yes, ...
S: If so, isn't it possible that in the future anyone
may discover a new regular solid?
T [to the class] What do you think?
Of course, the answer to the student's question is one
and only one: if the definition of regular solid is the one
used in the presented proof, there are no more than five
regular solids, so it is impossible to find another one. For
the students involved in the lesson the word "impossible"
sounded "too strong" and "very dramatic". It turned on a red
light on my head: How do they understand impossible
statements? How do they prove such statements?
To learn more about these questions, an
open questionnaire was designed and delivered to the
students a week later. They were asked to think about the
following questions and to write down their opinions.
1. What is the meaning of "impossible" in
mathematics?
2. Enunciate three examples of mathematic impossible
and correct
statements.
3. How do you explain to your math students that
something is impossible?
4. In your opinion, do we - as mathematics educators -
have to expose our students to mathematic statements
involving impossibility? If so, to what purposes? If not,
why?
5. Let us define a new concept: A Centrified Triangle
is a right-angle triangle whose circumcenter is also its
baricenter.
Enunciate some properties of the set of Centrified
Triangles.
6. Prove or refute the following statements:
i) It is impossible for a kite to have
exactly one right angle;
ii) It is impossible for a non special
parallelogram to be a cyclic quadrilateral;
iii) It is impossible to find 3 collinear points in
the same circle;
iv) It is impossible for a square to have a
diagonal whose length is a rational number;
v) It is impossible for a straight line which is
not tangent to a parabola to have only one common
point with the parabola;
vi) It is impossible for a function to be odd and
even at the same time.
I'd like the reader to think about these questions before
I present my analysis.
This study tries to examine students' understanding of
the notion of "impossible" and the ways in which they prove
or disprove impossibility statements. Instead of discussing
their answers to the questions, a profile of four students -
Abi, Bernie Carmen and Dalia - will be presented, according
to their responses to the first two questions. The other
questions will be used to give strength to the ideas.
Abi wrote that
for him impossible is "when there is no object that
fulfills the requirements". He developed his ideas saying
that he identified impossible with the empty set: "If each
one of the requirements is translated into the set of
objects that fulfill it, then the objects that belong to the
intersection set fulfill all the requirements. If this
intersection set is empty, then I call the situation
impossible." In his search for properties of the "Centrified
Triangles", Abi explicitly wrote that he identified the set
of triangles for which their baricenter is also their
circumcenter as the set of equilateral triangles. Then he
follows: "Since there is no triangle that is both
equilateral and rectangle, I conclude that the set of
Centrified Triangles is empty. So, it is impossible for a
triangle to be Centrified." The examples of impossibility
statements Abi produced are: " There is no triangle for
which a side is equal to the sum of the other two", "Since
the solution set of the equation sin x = 2 is empty, it is
impossible to find a real number x for which sin x = 2 ",
and "It is impossible to divide 5 by 0, since there is no
number x that fulfill the condition 0x = 5".
Bernie wrote
that impossible is "when you cannot obtain an answer
to the question, when you cannot find a way to solve a
problem, or when you cannot prove that something indeed
exists." It seems that for Bernie, impossibility is a
subjective property, meaning that some mathematical task may
be impossible for him but possible for a friend. One of the
examples he mentioned was Fermat Last Theorem. and he
explained his choice: "It was impossible for almost four
centuries, but now it is possible". It may be important to
point out that the impossibility Bernie wrote about is not
the same as the impossibility that Fermat himself wrote
about. Fermat wrote in Latin: "On the other hand, it is
impossible for a cube to be written as a sum of two cubes or
a fourth power to be written as a sum of two fourth powers
or, in general for any number which is a power greater than
the second to be written as a sum of two like powers. For
this I have discovered a truly wonderful proof, but the
margin [in his copy of Diaphantous' Arithmetica] is
too small to contain it." (as quoted by Young, 1992, p.42).
Fermat wrote about the non-existence of three integers that
fulfill certain requirements and Bernie was thinking about
the fact that the problem was open. It seems that this
student confused the terms "unsolved" and "unsolvable".
When Bernie was asked how to prove that a
result is impossible he wrote: "I don't know if you can do
that at all... You only prove positive statements. I think
you cannot prove that something is impossible. If you prove
something, then it may be, it may exist. So, I think it is
impossible to prove that something is impossible".
Carmen wrote that
- for her - impossible is "something that contradicts
mathematics laws, principles and definitions, something that
if you do it, it leads you to an absurd, something you
cannot do in a specific framework of definitions, axioms and
theorems." In Carmen words, very important aspects of the
impossibility in mathematics may be identified: the
contradiction to the structure built and the relativity of
the notion possibility-impossibility to the system of axioms
and definitions chosen. One of the examples of impossibility
she mentiones was "It is impossible to take the square root
of a negative number if you are talking about Real numbers.
If you think about the Complex Numbers, it is a different
story. The same idea is true if you think about other
operations defined in more simple sets. For example, you
cannot substract 10 from 7 if you are thinking of natural
numbers. This operation is impossible in N but possible in
Z". The other example she gave was no less interesting and
rich: "Let's think about two classical problems: Squaring
the Circle or Trisecting the Angle. These results are
impossible if the tools allowed are the Euclidean Tools, but
with other tools these problems are solved, for example you
may trisect any angle with the help of the trisectrix.
Dalia wrote that
it is impossible "To prove an axiom or to define a
fundamental concept". Her example does not belong to the
same category of the other mentioned examples. She wrote
about an impossibility statement of the language in which
mathematics is written, while the other examples are
theorems of mathematics (Davis, 1986, p.68). One of the
examples she presented was "It is impossible for two
parallel lines too meet" In this case, it seems she used the
definition of "parallel lines" and built a statement of the
form "It is not the case...". This algorithm to built an
impossibility statement in mathematics was frequent among
other students too.
This four cases are going to be the protagonists of the
session.
Reference
Davis P.J. (1986) When Mathematics Says No
Mathematics Magazine 59(2) 67-76
Young R.M. (1992) Excursions in Calculus - An
Interplay of the Continuous and the Discrete Dolciani
Mathematical Expositions Number 13, M.A.A.
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