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sessment approach (Siegler, 1986) may be considered as updates of
Piagetian theory within the language of the information-processing ap-
proach that shaped cognitive psychology in the late 1970s and 1980s.
REFERENCES

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THE INTERACTION BETWEEN THE FORMAL, THE AL-
GORITHMIC, AND THE INTUITIVE COMPONENTS IN A
MATHEMATICAL ACTIVITY
Efraim Fischbein
Tel Aviv
1. INTRODUCTION
Essentially speaking, mathematics should be considered from two points of
view: (a) mathematics as a formal, deductive rigorous body of knowledge
as exposed in treatises and high-level textbooks; (b) mathematics as a hu-
man activity.
The fact that the ideal of a mathematician is to obtain a strictly coherent,
logically structured body of knowledge does not exclude the necessity to
consider mathematics also as a creative process: As a matter of fact, we
want students to understand that mathematics is, essentially, a human activ-
ity, that mathematics is invented by human beings. The process of creating
mathematics implies moments of illumination, hesitation, acceptance, and
refutation; very often centuries of endeavors, successive corrections, and re-
finements. We want them to learn not only the formal, deductive sequence
of statements leading to a theorem but also to become able to produce, by
themselves, mathematical statements, to build the respective proofs, to eval-
uate not only formally but also intuitively the validity of mathematical
statements.
In their exceptional introductory treatise, "What is mathematics?"
Courant and Robbins have written:
Mathematics as an expression of the human mind reflects the active will, the con-
templative reason, and the desire for aesthetic perfection. Its basic elements are
logic and intuition, analysis and construction, generality and individuality.
Though different traditions may emphasize different aspects, it is only the inter-
play of these antithetic forces and the struggle for their synthesis that constitute
the life, the usefulness and supreme value of mathematical science. (Courant &
Robbins, 1941/1978, p. I).
In the present paper, I would like to consider the interaction between three
basic components of mathematics as a human activity: the formal, the al-
gorithmic, and the intuitive.
1. The formal aspect. This refers to axioms, definitions, theorems, and
proofs. The fact that all these represent the core of mathematics as a formal
R. Biehler, R. W. Scholz, R. Sträßer, B. Winkelmann (Eds.),
Didactics of Mathematics as a Scientific Discipline, 231-245.
© 1994 Dordrecht: Kluwer Academic Publishers. Printed in the Netherlands.

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FORMAL, ALGORITHMIC, AND INTUITIVE COMPONENTS
science does not imply that, when analyzing mathematics as a human pro-
cess, we may not take them into account.
Axioms, definitions, theorems, and proofs have to penetrate as active
components in the reasoning process. They have to be invented or learned,
organized, checked, and used actively by the student.
Understanding what rigor means in a hypothetic-deductive construction,
the feeling of coherence and consistency, the capacity to think proposi-
tionally, independently of practical constraints, are not spontaneous acquisi-
tions of the adolescent.
In Piagetian theory, all these capabilities are described as being related to
age – the formal operational period. As a matter of fact, they are no more
than open potentialities that only an adequate instructional process is able to
shape and transform into active mental realities.
2. The algorithmic component. It is a mere illusion to believe that by
knowing axioms, theorems, proofs, and definitions as they are exposed for-
mally in textbooks, one becomes able to solve mathematical problems.
Mathematical capabilities are also stored in the form of solving procedures,
theoretically justified, which have to be actively trained. There is a
widespread misconception according to which, in mathematics, if you un-
derstand a system of concepts, you spontaneously become able to use them
in solving the corresponding class of problems. We need skills and not only
understanding, and skills can be acquired only by practical, systematic
training. The reciprocal is also sometimes forgotten. Mathematical reason-
ing cannot be reduced to a system of solving procedures. The most complex
system of mental skills remains frozen and inactive when having to cope
with a nonstandard situation. The student has to be endowed with the formal
justification of the respective procedures. Moreover, solving procedures that
are not supported by a formal, explicit justification are forgotten sooner or
later.
Certainly, there is a problem of age, of the order of what to learn first and
how to teach. But, finally, I expect that students, who learn the basic arith-
metical operations, for instance, are taught sooner or later not only the al-
gorithms themselves but also why they do what they do. This profound
symbiosis between meaning and skills is a basic condition for productive,
efficient mathematical reasoning.
3. A third component of a productive mathematical reasoning is intuition:
intuitive cognition, intuitive understanding, intuitive solution.
An intuitive cognition is a kind of cognition that is accepted directly
without the feeling that any kind of justification is required. An intuitive
cognition is then characterized, first of all, by (apparent) self-evidence. We
accept as self-evident, statements like: "The whole is bigger than any of its