Book · January 994 citations 110 reads 2,264 authors

a ten-week study of twenty human mathematics tutors. Paper presented at the
International Psychology of Mathematics Education (PME) Conference, Assisi, Italy.
Lesh, R., & Lamon, S. (1992). Assessing authentic performance in school mathematics.
Washington, DC: AAAS.
Lesh, R., Post, T., & Behr, M. (1989) Proportional reasoning. In M. Behr & J. Hiebert
(Eds.), Number concepts and operations in the middle grades. Reston, VA: National
Council of Teachers of Mathematics.
Maher, C. A., Davis. R. B., & Alston, A. (1991). Brian's representation and development of
mathematical knowledge: A four-year study. The Journal of Mathematical Behavior,
10(2), 163-210.
286
omnipotence about what constitutes "true" learning, instruction, and as-
sessment, we remain open during the teaching experiments (and afterwards)
to corrections to and revisions of our own models of these concepts.
4. SUMMARY
In summary, constructivist perspectives on learning have radical conse-
quences for how we define knowledge for children and teachers and re-
searchers. Constructivism demands authenticity in instruction and assess-
ment and nonabsolutism in the design and interpretation of teaching exper-
iments. The knowledge gained from teaching experiments remains histori-
cal, situated, and open to revision since it involves the application of the
researchers' best current theoretical models to help understand the cognitive
models of teachers who, in turn, are growing in their understanding of the
cognitive models of their students.
REFERENCES

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CHAPTER 6
DIFFERENTIAL DIDACTICS
edited and introduced
by
Roland W. Scholz
Bielefeld / Zürich
Apparently, mathematics is learned by different populations that, at least
phenomenologically, show variations in mathematical performance and in
access to the acquisition of mathematical knowledge. The problem regard-
ing how this variation in performance and perhaps ability may be explained
is approached from different points of view. Two extreme positions may be
identified with respect to the impact of gender, socioeconomic status, social
and ethnic minorities, culture, and personality on the learning of mathemat-
ics,
One position assumes that there are no systematic fundamental differ-
ences between different groups, such as the genders, with respect to the
learning of mathematics. According to this view, in some respects, every-
body equals anybody like nobody is equal to someone else.
The other position postulates that there are systematic differences in the
structure and dynamics of gender, socioeconomic status, race, culture, and
personality with respect to learning mathematics. With reference to the sub-
discipline of differential psychology (Anastasi, 1954), the branch of re-
R. Biehler, R. W. Scholz, R. Sträßer, B. Winkelmann (Eds.),
Didactics of Mathematics as a Scientific Discipline, 287-290.
© 1994 Dordrecht: Kluwer Academic Publishers. Printed in the Netherlands.

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search on mathematics education that systematically investigates differences
with respect to these variables can be designated as differential didactics.
Clearly, the question as to which of the two positions is adequate is not
just theoretical but must be answered by empirical research. Today, both
positions can be found among researchers concerned with the possible im-
pact of differential variables.
Any research that runs under the heading "differential didactics" should
include:
1. a diagnosis encompassing a category or a variable (i.e., an independent
variable) and a hypothetical or an empirically accessible outcome (i.e., a
dependent variable) in order to describe what is different;
2. an explanation or model for the genesis of differences between groups;
3. a description of didactical efforts and their impact on the "outcomes" of
mathematics education.
With respect to knowledge of the state of the art presented in the contri-
butions of this chapter, some general conclusions for investigations of dif-
ferential didactics can be suggested. For instance:
INTRODUCTION TO CHAPTER 6
288
In most industrialized countries, about 5% of the population are currently
considered to be mathematically (mentally) disabled in that they are referred
to special schools where they receive a mathematics instruction that differs
from that of the rest of their age cohort. Conversely, a small percentage of
students has excellent access to mathematics.
The question whether highly gifted students or different genders should
be promoted in special programs of mathematics education belongs to a set
of "circularily reappearing" questions that are often dealt with controver-
sially. H. G. Steiner (1986, pp. 280-282) has pointed out two issues that are
important for a scientific understanding of debates on these topics: First,
these topics define political paradigms (Dubiel, 1985): Any position and ar-
gumentation is defined by conceptions of humanity, cultural values, or so-
cial-philosopical theories. Second, didactics of mathematics as a scientific
discipline has to rationalize these controversies by revealing the foundations
of the different positions in the scientific knowledge and facts already at-
tained.
1.
2.
3.
4.
Standard tests are no suitable means to provide insight into the genesis of
group differences in mathematical thinking and in learning mathematics.
Mathematical learning and thinking is a complex affair shaped by social
experience, "personal life formation," and adaptation to cultural con-
straints. As a result, it is unusual to find a single variable or a monocausal
explanation for these differences.
Presumably, many differences in mathematical performance are due to
different internal representations and qualities of information processing
(see chapter 5 on psychology of mathematical thinking).
Didactical efforts used to optimize mathematics education for different
populations should vary in quality and not in quantity.

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ROLAND W. SCHOLZ
Before I turn to the different papers, I want to point out a potential re-
search mistake, which I shall call "hidden type I error," that may be inherent
in any study on differential didactics.
The potential research mistake has been discussed nicely with respect to
sex differences in laterality by Springer and Deutsch (1981). For instance,
there are apparently more surveys reporting that male rats show signifi-
cantly thicker right hemispheres than female rats. Does this mean that this
finding is generally true? It is usually not known how many studies were
run on an issue. Furthermore, there is a lack of information on which studies
got published and which have remained unpublished. Thus there may be a
hidden type I error due to the scientific community's convention that only
significant results and/or findings that are related to hypotheses are pub-
lished.
Does mathematics learning show a similar bias to that suspected in later-
ality? Clearly we have to admit that we cannot answer this question and
hence we cannot exclude this possibility.