There exists the commonsense notion that mathematics should be taught because the study of mathematics, in some general way, improves people’s thinking “Correct- ness in thinking,” said Robert Maynard Hutchins , “may be more directly and impressively taught through mathematics than in any other way” . It is a notion with roots in the writings of Plato:

Those who are by nature good at calculation are, as one might say, naturally sharp in every other study, and …those who are slow at it, if they are educated and exercised in this study, nevertheless improve and become sharper than they were. [Republic, Book VII, Grube translation, 1974, p 178]

The same dialogue from the Republic includes the suggestion that “calculation” is “not to be neglected” because one could not “easily find studies which give more trouble to the practicing pupil”.Thisideaofthebenefitsof learning to overcome difficult obstacles also persists today In their Case Studies in Science Education, Stake and Eas ley summarized the views of teachers they inter- viewed as follows:

Teachers said they could not change the fact that life is going to require youngsters to do a lot of things that do not make sense at the time, and that seem very difficult. Therefore, they argued that there should be a significant body of learning at every grade level which is difficult, which may not make sense at the time, but which has to be learned by every student.

Mathematics is often thought of as a part of the “signifi- cant body of learning” mentioned by those teachers.

The idea that the study of mathematics improves one’s thinking and helps one learn to overcome difficult obsta- cles was part of the dominant curriculum theory ofthe 19th century, mental discipline theory. The role of mental disci- pline theory in the history of mathematics education is the focus of this article. More specifically, I will argue that we now remember mental discipline theory only as a carica- ture of what it was. It had its fanatic followers who believed it was best for children to study those subjects they most disliked, but it also had more sensible advocates who expressed a basic faith in human intelligence. The tum of the century is generally seen as the time when mental discipline theory was repudiated.. It did begin to decline in influence at this time, but arguments about it, particularly

For the mental disciplinarians, then, the mind was to be

viewed if it were a muscle the “as if’ quality of this

metapho and any other curriculum metaphor must be stressed. Through metaphor, we are asked to view one thing (what is to be explained) as if it were something else (that with which we are familiar). These two components of the metaphor are not to be taken as literally the same thing. The metaphor provides a lens through which we may see what we want to explain in another way. Like a lens, the metaphor emphasizes certain points and dimin- ishes (or even distorts) others If we recognize this lens-like quality, metaphor can be a valuable asset to our thinking It is crucial, however, to remember that metaphor, used in this way, not only helps but also shapes our perception; it “has imbedded in it an element ofpersuasion, and one who is not critically aware of the power of metaphor can easily become its victim” [Kliebard, 1982, p 15]

Mental discipline theory through the eyes of Char’les William Eliot A transitional figure in this era, was Charles William Eliot Eliot espoused the basic tenets of mental discipline theory, but he also attempted to adapt somewhat to the changes taking place in society and in the school

This idea that “each mind ..is its own method,” when combined with Hall’s concern for the different interests and probable destinations in life of particular individuals, essentially captures his view toward individualization of the cuniculum based on the natural order of development of the child

One must, however, be careful not to associate the idea of individualization exclusively with Hall and others like him.. Eliotalsoadvocatedaformofindividualization,arguing that it is a waste for society, and an outrage upon the

individual, to make a boy spend the years when he is most teachable in a discipline the end of which he can never reach, when he might have spent them in a dif!erent discipline, which would have been rewarded by achievement Herein lies the fundamental reason for options among school as well as college studies, all of which are liberal. A mental discipline which takes no account of differences of capacity and taste is not well directed. It follows that there must be variety in education instead of uniform prescription

No teacher of mathematics who believes that the study of his subject can contribute to the develop- ment of the pupil’s general intelligence and who teaches with such development as one of his aims, need be disquieted by any assertions that psychology is against him. Even though he may see no whit of improvement in the pupil’s mathematical thinking, or in his ability to handle the subject’s more complicated processes, or in his appreciation of its wide range of usefulness, if the teacher succeeds only in giving the pupil such increased familiarity with the most elementary notions of the subject as will enable him with greater speed and accuracy to mark a cross in the square and not the triangle, and to draw a line from the second circle above the third below the fourth to the fifth, he may rest content with the knowledge that the best psychologists that the Uni- ted States Government could summon to its aid will, in consequence of that improvement, rate that pupil higher in general intelligence.