TEACHING MATHEMATICS THROUGH FORMALISM: A FEW CAVEATS
Torkel Franzen
Swedish Institute of Computer Science
torkel@sics.se
There is much to be said for both the charm and the utility of
formal manipulation in reasoning. It is of course possible to
experience the utility without the charm, or conversely. As an
instance of the latter, T.S.Eliot has testified to the pleasure he
found in pushing symbols around when he was taught logic by Bertrand
Russell, although he did not connect the symbols with anything in
reality, while on the other hand innumerable students and engineers
have sweated through dreary calculations quite lacking in charm in
order to arrive at presumably useful results. In the happiest
applications, though, formal manipulation carries us gracefully to a
desirable goal, allowing us as if by magic to reach interesting
conclusions in a painless fashion. It is not surprising that
mathematicians as well as physicists have put great store and faith in
the power of formal manipulation.
To be sure, the bold leaps of people like Euler or Dirac, using old
formalisms in new ways with no guarantee that the results make sense,
will in most cases - when we don't have the peculiar talents of an
Euler or a Dirac - yield poor or nonsensical results. The chief use of
formalism must consist in sticking to rules known to be correct, and
this is what we try to teach students when we teach them to use
various formalisms: to stick to the rules, to let the rules guide
them, and to arrive at results formally but without drudgery.
In recent years, the book by David Gries and Fred B. Schneider, _A
Logical Approach to Discrete Math_, has forcefully presented the case
for consistently using an equational formalism in teaching logic and
discrete mathematics, and for stressing the role of formal
manipulation in general. This text has been used with good results,
and rather large claims have been made by David Gries and others for
the general efficacy and usefulness of this formal approach.
My purpose in the following is to present a somewhat dissenting
view. Without denying the usefulness and attraction of formalisms, I
maintain that we cannot draw any definite conclusions regarding the
efficacy of a formal approach to teaching logic and mathematics
without testing the degree of understanding exhibited by students in
contexts divorced from the classroom use of the equational formalism. It
is is very much open to debate whether "a solid understanding of what
constitutes a proof and a skill in developing, presenting, and reading
proofs" (quoting from the Instructor's manual for the book) is related
in any way to facility in formal manipulation, and also whether
"teaching rigor calls for carefully laying out rules for syntactic
manipulation".
It goes without saying - a phrase not infrequently accompanying the
saying of what goes without saying - that my comments in the following
are not in any sense authoritative. I have no particular expertise in
pedagogy and no profound insights to offer. Essentially, I'm just
chewing the fat about his book with David Gries and the rest of you
(unfortunately unknown to me, since I'm unable to attend the workshop),
in the somewhat blunt but I hope not abrasive style of my earlier
comments on a mailing list where I first learned about the book.
My skepticism is based on some very simple considerations. First,
testimonials aren't really a good measure of the degree of
understanding achieved by students. Enthusiastic and dedicated
teaching will always evoke a corresponding response, and naturally enough
many students will express themselves in enthusiastic terms about what
they have learned through such teaching, whatever the methods used.
For a more informative and objective measure of what has been
achieved, it is necessary to examine the students from a vantage point
that is independent of the particular pedagogical scheme in question.
And in the case of the equational logic, and general formal approach,
of Gries and Schneider, I cannot help but suspect that the theorems
and problems presented in the course have been chosen so as to admit a
painless treatment in the terms offered by that scheme. After all,
logical manipulation of formulas only takes us so far. In proving a
highly non-trivial combinatorial result such as Ramsey's theorem, we
are not helped at all by the manipulation of equational formulas, but
need, rather, to be able to follow inductive reasoning applied to a
complex combinatorial problem.
Ramsey's theorem is not in fact treated in the book, so I will base
my further comments - amplifying the general misgivings expressed
above - on a variant of a theorem put forward by David Gries as an
example of the efficacy of his approach:
THEOREM. (U,<) is well-founded iff the principle of
mathematical induction holds for (U,<).
Recall that a subset < of UxU is a well-founded relation iff every
non-empty subset M of U has at least one member x which is minimal with
respect to <, i.e. such that yV, there is a
unique function F:U->V satisfying
F(x)=H(x,F|{y:y rk(y)i". Suppose some Ai is true. Then Ai+1
is false, but then Ak is true for some k>i+1, so Ai is false after all.
So every Ai is false. But then Ai is also true!
This argument can be recast to prove our theorem in one direction,
assuming < to admit definitions by recursion, as follows. Suppose P is
a subset of U that has no minimal element, and define (by recursion
applied to the characteristic function of M):
x belongs to M if and only if for every y