TASK VARIABLES IN A
QUESTIONNAIRE ON PROBABILITY'S PRODUCT RULE
Ernesto Sánchez Sánchez
CINVESTAV-IPN. México.
Román Hernández Martínez
CINVESTAV-IPN. México.
The content core as defined by the product rule in probability —problems
which can be solved by means of the P(AÇB)=P(A)·P(B) formula, or by some
equivalent procedure— is a particularly difficult matter in its solving
procedure by the students. Following
the methodology suggested in Goldin’s (1984) compiled works, we have identified
and defined task variables concerning
the probabilistic product. The ability
to classify and to define task variables will allow both the researcher and the
teacher to have a systematic control to determine the effects of such variables
on the solving behaviour [Kulm, 1984, p. 1].
We have prepared a questionnaire consisting of six items that are
related to the concept of the product rule in probability where such rule can
be used to solve them, although a combinatory scheme and the classic
probability formula can also be employed.
We have regarded three main task variables: 1) time, refers to situations that can be of two kinds: a) synchronic, if two (or more) events
occur simultaneously, and b) diachronic,
if the events happen in succession; 2) 'distinguishability',
refers to the possibility of distinguishing —or not— objects from one another
(it has already pointed out that this 'small' difference propitiates confusions
in combinatory tasks [Batanero et al., 1994]); and 3) the amount (few, many) of objects considered —if they are but a few,
the solution is easier than if they become a big bunch. In constructing choices to each question, we
considered two wrong strategies,
which had been previously detected among high school students [Buendía, 1994]:
a) additive and b) ‘pseudoadditive’.
The problem in each item can be divided into two moments, and for each
of them the probability must be calculated —the correct answer is the product
of the probabilities. Many students
perceive the two moments and they calculate their respective
probabilities. However, not all of them
perform their product (the multiplicative strategy); some perform an addition
(the additive strategy); and there are some others who apparently perform an
incorrect addition of fractions, as if they were applying the following rule: 
(a pseudoadditive
strategy). This answer could be produced
by a model other than an incorrect addition.
The questionnaire was applied to 196 students from different levels and
two institutions. We analized the
results and made observations about performance, order of difficulty in the
items and the relationship among them considering de task variables.
References
Batanero, M. C.; Godino, J. D.;
Navarro-Pelayo, V. (1994). Razonamiento Combinatorio. Síntesis, España.
Buendía, G. (1994). Observaciones
acerca de las respuestas frente a tareas que involucran la regla del producto
probabilístico, MSc Dissertation, Departamento de Matemática Educativa del
CINVESTAV-IPN. México.
Goldin, G. A.; McClintock, C. E.
(1984). Task Variables in Mathematical Problem Solving. PA: The
Franklin Institute Press.
Kulm, G. (1984). The Classification of Problem-Solving Research Variables, in
Goldin/McClintock, Task Variables in mathematical problem solving (pp. 1‑21). PA: The Franklin Institute Press.