Capítulo 2. La intervención con adolescentes con conductas problemáticas y sus familias
2.6. Grupos multifamiliares con adolescentes con comportamientos problemáticos
The conviction that there may exist many limits of a function for x approaching one point x0 can be an example of how visionary mind work. Analysing the graphs the student described how he determined the limits of functions. At first he considered figure 1.
He thought in the following way: “Firstly, I look at x-es from −
∞
, then from +∞
. They do not have to belong to the domain. I approach 1 [he moves a pencil along the graph], until I reach the limit point. This end of the graph will be, let’s say 1 for x=−21, so lim ( )x→1 f x =1.
Then I approach 1 but there are no x-es from the domain here, only x0 =1. Since there are no x-es close to x0, from the domain 2 is not the limit for x approaching 1. It is only the limit at this point. Then I approach 1 from +
∞
and similarly obtain the limit for x approaching x0. It is lim ( )x→1 f x = 1 2
2 , because the limit point will be here, let’s say 21
2 for x=2 ”. It is worth
adding that the student when asked explained that “limits for x approaching −1
2 are also equal
to 1 and 21
2 but had the function not been defined at x =2 , 2 1
2 would not have been a limit”.
He considered numbers −1
2 and 2 as the ends of extreme intervals contained in the domain
and 1 and 21
2 as the values corresponding to them. Next he analysed figure 2.
figure 2
He used similar reasoning and in addition to limits connected with figure 1 he pointed at lim ( )
x→1 f x =2 . When asked about the points of the first coordinates which were the ends of the
interval containing x0 and defined by him as
(
1)
2 1 2 1 , and(
11 2)
2 1 3, , he concluded that “these are not the limit points and there are no limits because x-es belong to the interval surrounding x0. But they would be the limit points if there were gaps on the graph somewhere before x0”.
He meant a function whose domain was, for example, a set: (−∞,−21 ∪ 4 3 2 1, ∪ 4 5 1 6 1 , ∪ 115 1 1 2
, ∪ 〈 + ∞2, ) . Its limits besides those given with reference to graph presented in figure 2 and the aforementioned 11
2 and 2 1
3 should be also the function values for the arguments
equal to 3
4 and 1 1
5. Such expectations were connected with appearance of two new “limit
points”
(
( )
43)
4 3, f and(
11( )
1)
5 1 5, f . The student determined the limits of other functions likewise. On the basis of his statements it may be concluded that the student distinguished two concepts of limit of a function, namely: the aforementioned limit with x approaching x0 and
the limit at the point x0 – equal to the function value.
From the discussion it followed that the concept of limit for x approaching x0 was connected with the student’s understanding of the very phrase “x approaches x0”. He interpreted the phrase as follows: if x approaches −21, then it approaches also 1. This conviction together
with its degenerating consequences were also revealed by other examined persons. However, the conceptions expressed by the student during the research had all the properties characteristic of visionary minds. The character of the associations was so peculiar that it is difficult to believe that they were influenced by teaching the meaning of the limit of a
function, text-books or the very ‘nature’ of the notion. Hence, the reasons of such associations may be looked for in visionary properties of their authors’ minds.
Schematic mind
The way of thinking the schematists adopted may be best explained by a few various examples. Some of them thought that the terms of sequence should form a formula or should be obtained by means of a formula. Their conceptions, however, related to more situations than just associating a sequence with a formula used ‘to produce’ terms. They accepted arithmetical and geometrical sequences even if they could not give their formulas. In this case, for them, ‘the formula’ was type of sequence – arithmetical or geometrical. When they considered the graphs they accepted these which were similar to the ‘familiar’ sequences. The word ‘formula’ meant for them a scheme into which – in their opinion – a sequence could be fitted.
Schematists also looked for ‘familiar’ functions graphs considering other problems. On the basis of the information about those functions they determined their features – limits, continuity, and even derivative (see Przeniosło, 2001, p. 110; 2003a, p. 83; 2004, p. 128). In the case of two-part formulas some pupils and students reduced the condition connected with the equality of one-sided limits to the form: “If the function is defined using a brace bracket I calculate the limits at x0 from each formula; if the limits are the same the function has the limit at x0 but if they are different the limit does not exist”. This condition was applied to sequences, other functions, points x0 ∈R and those being plus or minus infinity; sometimes also in the case of the function defined by several formulas.
The similar condition was often used during considering continuity and derivative of the function defined with brace bracket. For example, considering continuity of the function f, where ∈ ∈ = dla , dla , 0 ) ( Q \ R x x Q x x
f , many schematists thought that f is continuous in the whole set of real numbers. They came to this conclusion in the following, schematic way: “The function f is continuous in R, because both functions y=0 and y=x are continuous. We have to yet consider continuity at the point x0 =0. The function is continuous at x0 =0, because lim
x→0x =
lim
x→00 = 0, that is the left-hand limit is equal to right-hand one and to f(0)”.