Arithmetic Classes Research Articles

Commercial Games for the Arithmetic Class

“Amusement and Pleasure ought to be combined with instruction in order to make the subject more interesting. There should be games of various kinds such as a game played with different kinds of coins mixed together. There should also be problems connected with boxing and wrestling matches. These things make a pupil useful to himself and more wide awake.”—Plato

Kurt Bing. On arithmetical classes not closed under direct union. Proceedings of the American Mathematical Society, vol. 6 (1955), pp. 836–846.

Kurt Bing. On arithmetical classes not closed under direct union. Proceedings of the American Mathematical Society, vol. 6 (1955), pp. 836–846.

Two concepts from the theory of models

In this note we point out that two concepts which have been introduced separately into the theory of models are extensionally equivalent. The basic ideas can be expressed conveniently in the terminology of Tarski's theory of arithmetical classes. It will be recalled that an arithmetical class is a class consisting of all algebraic structures which satisfy some fixed sentence of the first-order functional calculus. A more general concept is that of a quasi-arithmetical class, whose elements are all algebraic structures which satisfy some fixed set of first-order sentences.Now where H and K are any two classes of algebraic structures, Abraham Robinson has defined the relation H is persistent with respect to K to mean that (i) H∩K is non-empty, and (ii) whenever M ϵ H∩K, M′ is an extension of M, and M′ ϵ K, then M′ ϵ H. Concerning this notion Robinson has established the following theorem: If K is the class of abelian groups (or of commutative fields) and H is an arithmetical class which is persistent with respect to K, then H∩K contains a group (or field) which is finite. This leads us to the following definition.

The Elementary School Mathematics Library: A Selected Bibliography

Continual ohowth in understanding mathematical ideas is dependent upon a wide variety of experiences used to enrich and stimulate fmther interest and activity on the part of the learner. The elementary school library is an excellent source of quantitative ideas, described both pictorially and symbolically, that go beyond the systematic study materials used in most arithmetic classes.

On Arithmetical Classes Not Closed Under Direct Union

On Arithmetical Classes Not Closed Under Direct Union

What is Going on in Your School?: Why Pupils elect to Take Mathematics

For several semesters at Bosse High School, the students in many of the mathematics classes have been required to carry out a project on some subject of math as an individual and contribute his results to the class. These projects are used in arithmetic, algebra, and geometry classes as an extra credit incentive for the students. The list of topics is usually developed by teacher-pupil discussions, the students being encouraged to originate their own projects to fit in with what the class is studying. After the student has selected his topic, a conference is held with the teacher to discuss plans for developing the project. The projects are done by the student on his own time and must be completed by a specified time.

On arithmetical classes not closed under direct union

On arithmetical classes not closed under direct union

A proof of the compactness theorem for arithmetical classes

A proof of the compactness theorem for arithmetical classes

The Mechanical Analysis of Sediments from Thin-Section Data

In 1935 Krumbein published a method for correcting the moments of a thin-section size distribution to obtain those of the loose-grain size distribution. His method is limited in two respects: (1) it does not give the total distribution, and (2) it can be applied only if the measurements are grouped into equal arithmetic class intervals. It is often desirable to know the total size-frequency distribution as well as the moments of this distribution. Moreover, it is customary to plot the analyses on a logarithmic, rather than an arithmetic, scale, so that for comparative purposes it is necessary to have a set of correction factors that can be used with logarithmically grouped data. In this paper Krumbein's theory is extended to achieve these objectives. In the first section a method for reconstructing the total arithmetic distribution is presented. The second and third sections are devoted to the logarithmic case; in the former section the logarithmic bias corrections are derived, and in the latter the metho...

Verbalization and Learning in the Classroom

T HE function of verbalization in the classroom is a problem that has received little objective examination. This article records a study of an attempt to relate verbalization to the status of learning in an arithmetic class. It is recognized that this study involves but a small part of the larger problem and springs from meager beginnings. Starting with the simple observation, apparent in the average classroom, that youngsters like to talk about what they are doing, further observation indicates that youngsters frequently talk about what they are thinking. Dewey points out that verbalization has an emotional stimulus toward intellectual work and can as-

Arithmetic for the Twelfth Year

I will not go into detail concerning the teaching of a twelfth year arithmetic class, because you teachers know far more about that than I, but I will give you the development of the class in Ridgewood and its possibilities. What we are doing in Ridgewood is not new. Many schools throughout the country are giving the same work but under various titles. There are some who feel that our spade work in this branch of mathematics will be interesting to you. I believe the course fills a definite need in our school and community. The fact that the joint commission in 1940 suggested a social-economic arithmetic course for grade twelve or that the Commission for Postwar Plans in 1944 urged the high schools to include new and better courses for a large fraction of the school population, whose mathematical needs are not met in the traditional sequential courses, might be necessary to justify such a course in the eyes of some educators or college admittance officers, but not to me, nor to most of you. Even in a community such as Ridgewood, where 75% of the high school students plan to enter college, to contend that the sequential courses are the only ones of value and other courses may not be included in the curriculum is either due to pure stupidity or laziness. We either can't see beyond a few college requirements or knowing something else should be done for the students, we pass over the opportunity because of the work involved.

The Art of Teaching: An Attempt to Develop Number Sense

The above incident occurred in a Seventh Grade arithmetic class; while this may be considered an extreme case, yet it is astounding how little number sense has been developed in pupils after seven or eight years of work in mathematics.

Transfer of Learning in Simple Addition and Subtraction. I

In the precedifg article of this series' the chief matter of interest consisted in the effect of teaching 200 number combinations in addition and subtraction compared with that of teaching only 11o combinations. It was shown that the group of pupils who studied i io combinations knew at least as many combinations in toto as the group studying 200 combinations but that the latter group knew a slightly larger percentage of the combinations which had been left untaught than did the former group. The present article deals with the results of the experiment chiefly from the point of view of transfer. In any one of the four tests how many of the combinations which had not been taught up to the time of the administering of the test did all the pupils in the experiment know? This problem raises the question: To what extent did drill on certain taught combinations transfer to other untaught combinations? An answer to the latter question hinges in part on the number of combinations children learn outside the regular arithmetic class periods. For this reason, a table will be included showing how much two groups of children learned outside the periods of arithmetic instruction. One group (designated as Group A) comprised one hundred children who received no instruction in arithmetic for the last twelve of the seventeen weeks of the experiment; another group (designated as Group B) comprised eighty-six children who had no formal arithmetic instruction during the entire seventeen weeks. Table III shows the scores achieved by these two groups. The reader will receive a better impression of the situation by referring to Figure i. In this figure the average scores in addition

The Effectiveness of Directed Study

It is the purpose of this article to present the results obtained in a comparative study made of two eighth-grade arithmetic classes, one of which was conducted as a directed-study class and the other as an ordinary recitation class. The classes were of the same size and were made up of boys only. When the seventh-grade records in arithmetic were compared, it was found that there was a slight advantage in favor of the recitation class. This finding was further substantiated by the scores made on the National Intelligence Tests. A sixty-minute period was used for the directed-study class. Part of the time was given to reviewing the work of the preceding day, part to assigning new work, and the remainder to directing the study of the assignment made. There was some variation in the amount of time devoted to each of these procedures, but the time available for study was never less than twenty-five minutes. The class was divided into three groups on the basis of ability. The pupils in the inferior group were given work suited to their ability and essential as part of the course. Those in the middle group were given some work in addition to that given to the pupils in the inferior group. The superior group was given part of the work assigned to the inferior and the middle groups and, in addition, supplementary problems from other books. Home work was not expected. The work done each day was collected at the close of the period. Suggestive directions for the study of arithmetic were given to each pupil in mimeographed form. These directions were discussed, and frequent reference was made to them during the study period. This training in the application of study directions did much to build up proper study habits. The class period for the recitation group was forty minutes in length so as to conform more nearly to the average recitation period. Most of the period was used in reviewing the problems that had

A Measurement for Your Arithmetic Classes

A Measurement for Your Arithmetic Classes