Mathematical Situations Research Articles

Implications Arising out of Students' Errors

… After thirty years in the profession, a mathematics teacher looks at mathematics education from the vantage point of the errors which beginning Freshmen make repeatedly, offering an overview of the total mathematical situation and emphasizing what the errors are and why they are made.

Mathematics and its Applications in the Sciences

We have undertaken to discuss the nature of mathematical systems, the way in which they are discovered, and the uses to which they are put in the empirical sciences. The empirical sciences employ mathematical systems in framing final formulations, but adopt mathematical techniques long before reaching that stage. It is a prerequisite that the mathematics they employ has been developed separately and within its own domain. We shall return to the relation between mathematics and the empirical sciences before we are done, but in the meanwhile we shall be constrained to begin by discussing the situation in mathematics in isolation from any empirical considerations.

A SURVEY OF MATHEMATICAL EDUCATION

A SURVEY OF MATHEMATICAL EDUCATION

Subjectivism and Objectivism in the Social Sciences

Philosophizing about the social sciences involves an initial problem of denotation. Although the natural sciences are the scene of intramural disputes like those between proponents of quantum mechanics and relativity theory, no one doubts either what the natural sciences are or that they are sciences; and all of them may be said to use, broadly speaking, the same scientific method. But the case of the social sciences is different. It resembles somewhat the situation in mathematics where the intuitionists deny that non-constructible concepts or problems belong to mathematics; and even more the situation in philosophy where groups like the analysts and the existentialists not only do not speak the same language but do not even recognize one another's right to be called “philosophy.” Similarly, the disciplines which comprise the social sciences today differ from one another in basic ways, involving long-standing disputes over which ones have a right to the names of the several social sciences and what their proper methods are. Familiar examples of these disputes are the controversies between the classicists and institutionalists in economics, and between the analytical and sociological schools in jurispurdence. Consequently, it is impossible for philosophers, or social scientists philosophizing about their general field, to proceed to discuss “the” subject-matter or “the” method of “the” social sciences as if the phrases containing these definite articles were definite descriptions on whose objects there was general agreement.

Mathematics Recreations

Many teachers of mathematics believe that a mathematical recreation is a species of its own. In other words, not all mathematical situations may be turned into mathematical recreations. This department proposes to explode this misconception once and for all. Any mathematical situation, if appropriately treated, may become a mathematical recreation. While “exploding” the above misconception, this department will reveal some of the “secrets of the trade” so that any mathematics teacher may become a creator of mathematical recreations.

The President's Page

Among the many activities of the National Council there is one I wish to call to your attention at this time. You will be interested because it concerns every teacher of mathematics. All of us, as teachers, are working for the improvement of our teaching and, being professional, we give of our ideas and our efforts to help the mathematics situation in general. Realizing the futility of working alone on our problems, we seek the cooperation of others and join one or more mathematics organizations. We do this with the hope of getting help on our teaching problems and to gain some inspiration for the job we are trying to do.

The Need for Extending Arithmetical Learnings

The current trend toward teaching arithmetic and mathematics for “meaning” and “understanding” coupled with the aim to achieve functional competence on the part of the pupil is causing reverberation among mathematics teachers. Teachers ask, “What and how much can pupils really understand?” and “Shall I teach this topic if I cannot find more than a few poor examples of functional usefulness?” While these questions do not represent the point of view of all teachers, they are indicative of a growing discontent with the achievement in mathematics at both the elementary and secondary school levels. Of course, people in business and industry, as well as representatives of the armed services in the late war, have repeatedly asked that curriculum workers and mathematics teachers do something (almost anything) that will educate the common citizen so that he has at least a modicum of competence in the mathematical situations with which he is repeatedly confronted.

The Present Situation in Secondary Mathematics, with Particular Reference to the New National Reports on the Place of Mathematics in Education

Introduction. The subject chosen for this discussion presents so many different aspects that it seemed necessary to agree at the outset on certain limitations of its potential scope.

The Introduction of Negative Numbers

It Is one of the most difficult tasks in teaching elementary mathematics in high schools to be both, understandable and correct. Most textbook authors know how to write simply and so to speak popularly, but, on the other hand, there is no doubt that many of them fall short of perfect scientific correctness. It would be ridiculous to try putting in school classes such modern scientific methods as, for example, the formal introduction of real fractions as couples of integral numbers, negative numbers as couples of positive numbers, imaginary numbers as couples of real numbers. However, we are often forced as teachers to meet a mathematical situation where we have to pay attention to two points: first, no conclusion must be gained surreptitously or by tricks, and secondly, when some of the steps within a statement are too difficult to be understood by young people the gaps must not be concealed but, on the contrary, pointed out to the students with the explanation that it is possible to accomplish the proof although we may not do so for the moment.

Some Comments on the Secondary Mathematics Situation

Some Comments on the Secondary Mathematics Situation

Thither Algebra

The Recent Article by Mr. Betz is a classic of timeliness and wisdom. It arouses the faintly breathing hope that something may yet be done. Mark Twain's remark about the weather might well be applied to the situation in mathematics. There is still however an observation which is perhaps worth making. Our splendid physical prosperity must have some educational basis which is partially responsible for it. Now surely if there is a weakness in our present national development it is spiritual rather than material, it is in culture rather than in mechanical advances. So we are compelled to ask those who raise the cry that we have not taught mathematics efficiently enough just what they have a right to expect. They have set up certain expectations for various stages of mathematical development in the child and have been disappointed to test and find their expectations are not met. Perhaps they were optimistic rather than wise. The technical schools may claim all the credit but it still leaves nothing to complain about as far as the mechanical side of our celebrated prosperity is concerned. The thing that we need most to be concerned about is whether those that succeeded in mathematics furnished a smaller per cent of bootleggers and divorcees than those who failed. I have no statistics to offer but I have always pursued my work as a teacher of elementary algebra and geometry encouraged by the hope that if my students were led to think straighter in mathematics they would have a greater chance of leading useful happy lives. No one has come right out and said that the faulty teaching of algebra and the rest of the debatable high school subjects were responsible for the increase in crime except perhaps Henry Ford with his announcement of millions for education. His proposals do not sound feasible enough as a general solution to warrant our sitting back and waiting for his specifications for the reforming of our high schools.

Generalized Lagrange problems in the calculus of variations

In the new dynamical theory of economics there arises a very general problem which can be said to be a generalization of the Lagrange problem in the calculus of variations.4 It will not be necessary to consider the formulation of the corresponding economic theory here since I have already done this in another paper.? It would hardly be fair, however, to introduce the reader to a rather unusual mathematical situation without giving some hint as to its origin. It seems desirable, therefore, to give first a brief economic formulation of the problem whose mathematical aspects will be discussed in this paper. If there are two producers of an identical commodity C, manufacturing, respectively, amounts ul(x) and u2(x) of C per unit time, subject to the respective cost functions 41(ul, ui', U2, u1, U3, u f, x) and 42(ul, ul', U2, UI', U3, U3 , X), where u3(x) is the selling price of C at a time x, then the respective profits obtained during an interval of time xo < x < xi are

Reconstructed Mathematics in the High School

Few are satisfied with the present mathematics situation in the high school, particularly in the first two years of the high school. Dissatisfaction is found in the college faculties which deal with the product, among the mathematicians who are looking for a foundation for productive scholarship, among the teachers who are looking for something better; and dissatisfaction coupled with ridicule is found among the business men, engineers, and others, who expect mathematics learned in the school to function in the practical affairs of life. The purpose of this paper is an attempt to analyze the situation, to find out what is the matter with mathematics in the high school, and if possible to throw some light on the way out.