Abstract
To what extent were the individual learning processes which we observed and described in the previous chapter determined by the instruction given? Which qualities of these learning processes are concerned here? And which aspects of the course need improvement?
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Notes Chapter 6
These general matters could also be regarded and analysed in terms of reflection. The change of course following a dilemma in the solution procedure (caused by getting stuck due to insurmountable numerical obstacles, for instance), which is characterized by transferring to another solution level or else by a renewed approach on the same level while avoiding the obstacles is an example of a change of standpoint, called ‘directed shifting’ by Freudenthal. The intended goal solving the problem is not lost sight of during this change of course. It would be interesting to analyse in more detail all the individual learning processes with regard to the phenomenon of reflection. On this point, too, we must however set ourselves limitations (see chapter 2, § 2.4 and note 31). See, in connection with this matter, also chapter 8 and Freudenthal (1978) (1979b) (1979c), Streefland (1980a) (1980c). Nelissen (1987) contains an extensive bibliography of literature concerning metacognition and reflection. In the description of his research results, however, he chooses a different (arbitrary) level classification of reflection than one based on an analysis of types of’ shifting’.
With reference to chapter 5 in connection with incorrect reasoning when comparing ratios we will suffice here by mentioning a few recent publications, namely, Hart (1985), Hasemann (1986a) (1986b). Moreover, the literature in general still contains an examination of this matter connected to the subject, that is, incorrect additive reasoning with ratios is not theoretically linked to N-distractor errors with fractions. An exception here is Noelting (1980), who states, keeping in mind the learning of fractions and ratios, that a student’s operational ideas concerning natural numbers should have to undergo a process of ‘adaptive restructuring’.
Hasemann(1986a) observed this phenomenon as well. For Benny, too (chapter 1), results may also have been method-dependent
This graphic representation was partly inspired by the work of Teule-Sensacq and Vinrich (1982). See also Streefland (1986f).
This is a matter of spontaneous differentiation. That this leads to ‘(self) determination’, as stated by Terwel (1986) is not much more than a bromide, because one can arrive at this conclusion de facto with any curriculum for all forms of organization. And it ignores, moreover, the fact that the student’s determination aptitude is also decisive for the manner in which he finds his way in a system in which spontaneous or learning process differentiation is applied.
See, in connection with this, Streefland (1985c).
Research into progressive schematization using the ratio table in Streefland (1982c) indicates this.
See, in connection with this, Streefland (1986a).
See Streefland (1985a) (1985c).
See Streefland (1982c).
Streefland (1982c).
This can, depending on the context of the learning process and the (early) phase in which it takes place, be termed reflection. The change of standpoint in question here is one of ‘parallel shifting’, to use Freudenthal’s term. See also note 1 in this chapter.
The phenomenon of a context or situation model is virtually only recognized within the realistic concept of mathematics education. This can be explained by the value this concept places on both horizontal and vertical mathematization. In no other theory is this the case. Precisely in horizontal and vertical mathematization do situation models form not only a connecting link between both forms of mathematization, but play as well a decisive role in the long run. See, in connection with this, Treffers (1987a), Streefland (1986b) (1987). As far as we know, only the American Lesh has distinguished situation models. See Lesh, 1984.
Lesh et al (1983) also attached a great deal of value to this aspect of concept acquisition.
These are cognitive process models as referred to by Greeno (1976). They lack, however, the artificial quality that characterized his models for producing equivalent fractions. See also Treffers and Goffree (1985) and Streefland (1986b).
For the matter of the rich context we refer to Treffers (1987a) and de Lange (1987; § 2.3). The latter’s term for this is contexts of the third class/rank/degree.?
These circumstances make it possible for monographs to function as one fruitful access to algebra. See, in connection with this, also Streefland (1987b).
A number of practical obstacles also stood in the way of the research being able to follow an optimal course. For example: the computer program was not ready on time, the research had to sometimes be interrupted for long periods, it was not possible to group a number of lessons on successive days. Moreover, we are dealing with development research in which one simply cannot determine in advance the course of the educational process.
See, in connection with this, Treffers (1987a). Confrey (1985) (1987), a declared constructivist, also emphasized the role that reflection can play in the constructive manner of learning mathematics. Freudenthal (1979b) argued that the link between constructing and proving in mathematics is forged by reflecting upon what has been constructed.
See Streefland (1987b).
See, in connection with this, Freudenthal (1987), Streefland (1985b) and Treffers (1987a). De Lange (1987; loc. cit. p.44) states, in connection with this: ‘A division of the clusters of activities of mathematization into two distinct components is rather arbitrary, to say the least … But a bipartition in a descriptive sense can be useful, not only to describe mathematization more clearly in concrete examples, but also to discriminate between different methodologies,…’ De Lange was regarding mathematization primarily on a local level, within the solution of independent, complex problems.
See, in connection with this, Lesh (1985), Streefland (1986f) and Treffers (1987a). The matter of instability in the learning process can also be regarded in terms of an impulsive or reflective attitude, as did, for instance, Radatz (1976). We feel, moreover, that it makes sense to regard this matter from the standpoint of individual learning processes in a mathematical-didactic context.
Freudenthal has repeatedly emphasized the importance for the learning process of keeping the sources of insight open.
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© 1991 Springer Science+Business Media Dordrecht
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Streefland, L. (1991). Internal Evaluation of the Learning Process. In: Fractions in Realistic Mathematics Education. Mathematics Education Library, vol 8. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3168-1_7
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